"On the Notion of Cause," pages 180-208
[Posted on the fiftieth anniversary of Bertrand Russell's death, February 2, 2020.] This chapter addresses some confusions around the notion of “cause” as used by philosophers – confusions that are substantial enough that it would be best if the term were avoided. The misunderstanding of causality leads to other mistaken notions. And though philosophers are confused about the “law of causation,” what does science actually employ in its stead?
Many sciences never talk of causation, even though philosophers assert its fundamental nature to science. But perhaps the lack of talk about causes in physics, for instance, isn’t dereliction of duty, but recognition of the absence of causes. The law of causation is outdated, and survives “like the monarchy, only because it is erroneously believed to do no harm [p. 180].”
Russell examines dictionary definitions of “causality” and related words, leading to the distinction between a propositional function (“x is a number”) and a proposition (3 is a number). Propositions are either true or false, but propositional functions can sometimes be true (for some values of x, say), and sometimes false. This line of reasoning suggests that something is necessary if it is the “predicate” of a propositional function, which means that the statement is true for all possible values of its argument.
Eventually, Russell avers that causality (event e2 is caused by event e1, say) is when, for any event e1 that occurs, within some interval of time τ, event e2 occurs. (Variations of this definition have been provided by John Stuart Mill and by Henri Bergson.) Some alternative definitions suffer from circularity or from not limiting the time between cause and effect.
To examine the definition and explore its applicability or inapplicability to science, Russell puts aside (for now) the problem of multiple causes, to focus instead on the meaning of “event,” and on the length of time separating cause from effect.
If we define an event too narrowly, then we will never again see its exact match in the future, and so the notion of causation would lose all purchase. “An ‘event,’ then, is a universal defined sufficiently widely to admit of many particular occurrences in time being instances of it [p. 187].”
The effect cannot take place at the same instant of time, nor immediately afterwards, thanks to the lack of infinitesimal time intervals – so, there must be some finite passage of time twixt cause and effect. As soon as we allow for such a time lapse, however, it becomes possible that in that interval, an intervening event (a meteorite strikes?) changes conditions sufficiently that the proposed “effect” will not occur. But this in turn means that the proposed cause is not itself sufficient to ensure the effect. Were we to start adding the state of the environment into our cause, we end up draining our cause-and-effect claim of any applicability, through the previously noted problem with excessive narrowness.
Despite our difficulties with explicitly expressing the logic of cause and effect, in our quotidian lives, there are many reliable “regularities of sequence [p. 187].” Some of them may be completely dependable, and even less dependable correlations can spur scientific advances. But science pursues or requires no “law” of cause and effect, it does not assume that there is an invariant causal relationship out there waiting to be discovered. As sciences advance, our old causal claims become more nuanced, with finer partitions of antecedents and consequences. When the antecedents are finely enough delineated that we are certain of the consequences, they are also fine enough that they will not occur again, that there is no predictive value to the causal relationship.
Philosophy has been riddled with misconceptions around causation. One is that causes and effects have to be similar, so that mental processes, for instance, could not arise from inert matter alone. Another [Russell enumerates five – RBR] is connected to free will, the notion that causes cannot make someone do something that they do not desire to do. But these desires themselves might be “caused,” even if there is free will in the sense of only doing what is desired.
What is left when philosophy abandons any putative law of cause and effect? Accept for the nonce the logic of induction, that when we have a long series of cases where a “cause” is followed by an “effect,” then this relationship is quite likely to hold in similar future cases. These series that spark induction, however, speak to likelihoods, not necessary consequences, with respect to future observations. The inductive cause-and-effect might lead us to claim that striking a match causes the match to ignite, but we will find that striking a wet match will still not do so. Nor does the inductive approach suggest that every event has some cause. And it seems to be overly inclusive, supporting the claim that night causes day, but we will not back away from accepting that in the sense we are now discussing, night does cause day. The rules we get from the inductive approach can always fail in the next observation, without violating any scientific law.
As science progresses, we tend to move away from cause and effect claims. Gravity involves rules that masses follow, but we cannot isolate one aspect of gravitational forces as causes and others as effects. The correct formulation is mathematical, a stability in the differential equations that characterize the system. But this is hardly some a priori rule of science, as some philosophers expect from their “law of cause and effect.”
Note also that in physics, the complete state at an instant not only determines the future, but the past. There is no temporal priority that makes earlier things causes of later things, and not the other way around.
While science does not stipulate any law of causality, it does accept, in the background, a certain uniformity of nature. Functions that have characterized relationships in the past are expected (inductively) to hold in the future, and if they do not hold, there is some more general law, capturing both sets of data, that does hold. As with other inductive arguments, this uniformity assertion is only likely, not certain, to hold. And if it prove false with respect to some scientific “law,” the rest of science is not thereby invalidated.
The effects of gravitation within the solar system depend, though minutely, on matter that lies outside of the solar system. As we have little knowledge of that matter, we cannot fully verify gravitational theory. But we can be very confident of our claims about gravity within the solar system anyway, irrespective of what is going on in the rest of the universe. The solar system, with respect to gravitation and over a given time period, is a “‘relatively isolated system [p. 197],’” where it will behave in a uniform manner (approximately) during that time period, no matter what is happening elsewhere. A system is “’practically isolated [p. 198]’” if, though there might be some outside situations that would lead relative isolation to fail, we rightly suspect that those situations do not arise. When it comes to falling bodies, the earth is relatively isolated, but it is not isolated with respect to tides.
It is evidence, not a priori reasoning, that leads us to believe some systems are relatively or practically isolated. “The case where one event A is said to ‘cause’ another event B, which philosophers take as fundamental, is really only the most simplified instance of a practically isolated system [p. 198].” The fact that A is always followed by B is true thanks to the relative unimportance, in this case and for this time, of what is going on in the rest of the universe. The (unknown) laws of the universe could still hold but produce instances where A is not followed by B.
Causality is the reed upon which people make out-of-sample inferences. If these inferences are legitimate, the system is deterministic. A non-deterministic system is capricious (p. 199).
Brains are part of the universe. It seems that there is a one-to-one mapping between states of one person’s brain and states of the universe. Assume also that there is a one-to-one relationship between the state of a mind and the state of the corresponding brain. This leads to the notion that there is a one-to-one mapping between a mind and the state of the universe. Whatever (sub)set of states determines the universe, then, we can find the same number of states of one person’s mind that also “determines” the universe. Those who are concerned that mind is determined by matter (outside of the brain) should recognize that the determination goes both ways.
Even if there are multiple states of mind for any state of the brain [Russell, page 203, cites Bergson for this claim], people are not thereby forced (by the world of matter) to take actions that they do not desire to take. A similar point applies to whether or not the universe is goal-directed (teleological); the answer to this query is independent of whether or not the world is mechanical, deterministic given the precise state of matter.
It is believed that our desires cannot alter the past but can alter the future. But this belief is a relic of our memory acting only in the backwards direction (and because generally we only have desires for things unknown to us): we could equally say that the future cannot be altered by our wishes (and were our wishes different the past would have been different).
Most supporters of determinism go beyond claims that whatever may be may be – they regard the world as being determined function-like, a function of earlier data. But with no constraints on the complexity of the function, such determinism is surely the case: at the extreme, the full data themselves can be mapped trivially into “functions” that describe their state at a given time. But what science seeks is the simplest formula consistent with the known facts, from within the infinite set of possible, and so far unfalsified, formulae.
Perhaps what science really seeks is formulae where time (in an absolute sense – as opposed to lapses of time) does not enter as an independent variable, and hence those formulae are uniform in the sense that they hold at any time.
What of free will? Surely the known facts suggest that some of our volitions are determined. But at this point, we cannot be sure that all of our volitions are determined (except in the trivial sense noted above). Nonetheless, our sense of freedom is irrelevant to the scientific question of free will. “The view that it has a bearing rests upon the belief that causes compel their effects, or that nature enforces obedience to its laws as governments do [p. 206].” Our sense of freedom in our volitions is consistent with an appropriate view of determinism.
If our will is determined, is it determined in the mechanical sense, where data concerning material elements are sufficient to generate our will? If our will is so determined, we still need not view this as the triumph of matter over mind: a system with a set of material determinants can also have a set of mental determinants. Nor does a deterministic system require some uncomfortable notion of necessity, where, for instance, we must act against our wishes.
Russell (pages 207-208) offers a one-paragraph summary of this chapter. What philosophers think of as a law of cause and effect is mistaken. Systems can have multiple sets of determinants. Free will may or may not exist, but in any event, it does not have to be in opposition to determinism.
Sunday, February 2, 2020
Wednesday, August 28, 2019
Mysticism and Logic, Chapter VIIIb
“The Relation of Sense-Data to Physics,” pages 164-179
Recall that this second post covering Chapter VIII is devoted to the following sections:
IX. The Definition of Matter (p. 164)
X. Time (p. 167)
XI. The Persistence of Things and Matter (p. 169)
XII. Illusions, Hallucinations, and Dreams (p. 173).
We still have the issue of what constitutes “matter,” even after we define a thing as the class of its various appearances. Individual appearances are affected by this “matter.” Appearances provide more detail as they become closer; so, we can characterize a thing’s matter as the limit of its appearances as the closeness of the thing goes to zero (p. 165). (We cannot be sure that empirically, such a limit exists; but, we can infer (with error) a limit from the appearances that we do observe.)
“The appearance of a thing in a given perspective is a function of the matter composing the thing and of intervening matter [p. 165].” The intervening matter, for instance, might be a mist or some element of the receiving sense-organ. As we get closer to the thing, the intervening matter is less relevant – hence the thing itself is the limit of appearances as we approach the thing in question. Appearances are deceptive, but the closer they are, the more confidence we have in them. Matter is not “more real” than sense-data, but it is more reliable – more substantive? – than any one piece of sense data. Of course, as we approach an object, we see that it is not one object, but several, and objects seem to be infinitely divisible in this fashion: a single appearance can include many “things.”
For a single observer (not more generally, as relativity makes clear), two perspectives can be ordered in time, with one perspective before, after, or simultaneous with the other. We can extend this notion to sensibilia, so that a “biography” is “everything that is (directly) earlier or later than, or simultaneous with, a given ‘sensibile’ [p. 167].” The world’s history, then, is comprised of the union of “mutually exclusive biographies.” [Russell cites A. A. Robb as a source for his own views on time. In 1925, Russell went on to write ABC of Relativity, which was the first Russell book I [Bert] ever came across and read. An amazing audio version of ABC of Relativity, read by Derek Jacobi, is available for free here.]
How can the time in different biographies be synchronized? In the case of a sound, we can’t say that in every biography containing the sound, it occurs simultaneously, because some listeners (those closer to the source) do hear it earlier. Russell chooses to build a “velocity of sound” into his framework for such audible sensibilia. A similar approach, using a velocity of light, is taken for visual sensibilia. This solution has the (perhaps unfortunate?) implication that in trying to encapsulate a thing at a certain instance, we use appearances that themselves do not all occur at that same instant. “The” time at which the thing is in a certain state is the lower limit of all these instances.
So, we now have correlated appearances (in various perspectives) that give us one thing at a specific instant. But what about persistence, the connection of that same thing at different instances? Again, we have to construct persistence, just as we constructed the notion of a thing at a certain time with different observers – now the construction involves observers at different times.
If we look at just a single biography, how do we know that the same thing exists at different moments? The different appearances must be correlated, display a type of continuity. But we don’t observe anything continuously, so our assumption of continuity is only a hypothesis – though a hypothesis that we already have employed in developing our notions of particulars from sensibilia.
Continuity doesn’t imply a constant material source. A homogeneous fluid like a sea possesses continuity, but the motions of sea water “cannot be inferred from direct sensible observation together with the assumption of continuity [p. 171].” So, for those seemingly persistent objects that we take to be collections of sensibilia, we must also include adherence to the laws of dynamics.
Motion is the change over time in those assembled sensibilia that we take to be the same object. Because we have some discretion over what sensibilia constitute the same object over time, our notion of motion has some unavoidable ambiguity, if continuity is our only guide. Requiring adherence to the laws of dynamics resolves this ambiguity – and we will assume that there is a unique grouping of appearances into things that possesses this coherence (p. 173). By this method, we can identify how appearances at different times can be attributed to the same object.
Much of the luster of physics is due to its empirical successes, despite our inability to generate appropriate sense-data to test some of the hypotheses of physics. Nonetheless, we find that the sense-data we do have are not in contradiction with those hypotheses, and indeed, the hypotheses, combined with some sense-data, allow us to predict other sense-data.
Russell concludes section XI (“The Persistence of Things and Matter”) with a definition and a claim. The definition is: “Physical things are those series of appearances whose matter obeys the laws of physics [p. 173].” The claim is that we know empirically that such things exist, and this knowledge completes the program, verifies physics, deduces stuff from sense-data.
But what about “unreal” sense-data like hallucinations, which are not correlated in the usual way with actual stuff? What if we dream about an unreal thing? Recall that our definition of stuff like a table is that it is the collection of all sensibilia that contain it. Given that definition, then, “as well sleeping, as waking,” any observer doesn’t sense the table, but only one sensibilia.
Dream objects exist in the private space of the dreamer; they lose the correlation with other private spaces that real objects possess.
Such notions as “existence” and “non-existence” do not apply to sense-data, though they can apply to things described in terms of such data. The fact that a sense-datum exists, and that that datum is of a table, does not imply that a table exists. [Russell cites the fuller, symbol-based discussion of this point in Principia Mathematica, and also makes reference (p. 176) to the theory of descriptions.]
Russell goes on (pages 176-179) to indicate how his theoretical development answers four common arguments against realism:
(1) The same object can appear differently to different people. Russell allows different perspectives, which render these observational differences to be irrelevant for the question of object reality.
(2) The same object can give us seemingly incompatible sense-data. A stick in water can appear bent to the eye, though straight to the touch. But when a stick is a collection of sensibilia, and we recognize that the proper inference from one sensibile to another need not be fixed, incompatible sense-data are not a problem for a realist view of matter.
(3) Objects in dreams often are considered to be unreal—but the sense-data that lead to dreams are as real as any. As noted, dream sense-data lack the usual continuity and correlation with other sense-data. Nonetheless, they are physical, subject matter for physics to deal with.
(4) Hallucinations are like dreams, but it takes multiple observers for it to become clear that one person is suffering from hallucinations. (Dreams would have the same property if they were more closely connected to a person’s sense-data when awake.) The person suffering from hallucinations is in no position to know whether this is the case, or whether other people are conspiring against him.
“From the above instances it would appear that abnormal sense-data, of the kind which we regard as deceptive, have intrinsically just the same status as any others, but differ as regards their correlations or causal connections with other ‘sensibilia’ and with ‘things’ [p. 179].” The problem is not unreal data but rather, our unjustified expectations. As a result, abnormal sense-data do not present an impregnable obstacle for the science of physics. The approach provided above, though quite preliminary, especially as regards its role for time, permits physics to be empirically verifiable.
Recall that this second post covering Chapter VIII is devoted to the following sections:
IX. The Definition of Matter (p. 164)
X. Time (p. 167)
XI. The Persistence of Things and Matter (p. 169)
XII. Illusions, Hallucinations, and Dreams (p. 173).
We still have the issue of what constitutes “matter,” even after we define a thing as the class of its various appearances. Individual appearances are affected by this “matter.” Appearances provide more detail as they become closer; so, we can characterize a thing’s matter as the limit of its appearances as the closeness of the thing goes to zero (p. 165). (We cannot be sure that empirically, such a limit exists; but, we can infer (with error) a limit from the appearances that we do observe.)
“The appearance of a thing in a given perspective is a function of the matter composing the thing and of intervening matter [p. 165].” The intervening matter, for instance, might be a mist or some element of the receiving sense-organ. As we get closer to the thing, the intervening matter is less relevant – hence the thing itself is the limit of appearances as we approach the thing in question. Appearances are deceptive, but the closer they are, the more confidence we have in them. Matter is not “more real” than sense-data, but it is more reliable – more substantive? – than any one piece of sense data. Of course, as we approach an object, we see that it is not one object, but several, and objects seem to be infinitely divisible in this fashion: a single appearance can include many “things.”
For a single observer (not more generally, as relativity makes clear), two perspectives can be ordered in time, with one perspective before, after, or simultaneous with the other. We can extend this notion to sensibilia, so that a “biography” is “everything that is (directly) earlier or later than, or simultaneous with, a given ‘sensibile’ [p. 167].” The world’s history, then, is comprised of the union of “mutually exclusive biographies.” [Russell cites A. A. Robb as a source for his own views on time. In 1925, Russell went on to write ABC of Relativity, which was the first Russell book I [Bert] ever came across and read. An amazing audio version of ABC of Relativity, read by Derek Jacobi, is available for free here.]
How can the time in different biographies be synchronized? In the case of a sound, we can’t say that in every biography containing the sound, it occurs simultaneously, because some listeners (those closer to the source) do hear it earlier. Russell chooses to build a “velocity of sound” into his framework for such audible sensibilia. A similar approach, using a velocity of light, is taken for visual sensibilia. This solution has the (perhaps unfortunate?) implication that in trying to encapsulate a thing at a certain instance, we use appearances that themselves do not all occur at that same instant. “The” time at which the thing is in a certain state is the lower limit of all these instances.
So, we now have correlated appearances (in various perspectives) that give us one thing at a specific instant. But what about persistence, the connection of that same thing at different instances? Again, we have to construct persistence, just as we constructed the notion of a thing at a certain time with different observers – now the construction involves observers at different times.
If we look at just a single biography, how do we know that the same thing exists at different moments? The different appearances must be correlated, display a type of continuity. But we don’t observe anything continuously, so our assumption of continuity is only a hypothesis – though a hypothesis that we already have employed in developing our notions of particulars from sensibilia.
Continuity doesn’t imply a constant material source. A homogeneous fluid like a sea possesses continuity, but the motions of sea water “cannot be inferred from direct sensible observation together with the assumption of continuity [p. 171].” So, for those seemingly persistent objects that we take to be collections of sensibilia, we must also include adherence to the laws of dynamics.
Motion is the change over time in those assembled sensibilia that we take to be the same object. Because we have some discretion over what sensibilia constitute the same object over time, our notion of motion has some unavoidable ambiguity, if continuity is our only guide. Requiring adherence to the laws of dynamics resolves this ambiguity – and we will assume that there is a unique grouping of appearances into things that possesses this coherence (p. 173). By this method, we can identify how appearances at different times can be attributed to the same object.
Much of the luster of physics is due to its empirical successes, despite our inability to generate appropriate sense-data to test some of the hypotheses of physics. Nonetheless, we find that the sense-data we do have are not in contradiction with those hypotheses, and indeed, the hypotheses, combined with some sense-data, allow us to predict other sense-data.
Russell concludes section XI (“The Persistence of Things and Matter”) with a definition and a claim. The definition is: “Physical things are those series of appearances whose matter obeys the laws of physics [p. 173].” The claim is that we know empirically that such things exist, and this knowledge completes the program, verifies physics, deduces stuff from sense-data.
But what about “unreal” sense-data like hallucinations, which are not correlated in the usual way with actual stuff? What if we dream about an unreal thing? Recall that our definition of stuff like a table is that it is the collection of all sensibilia that contain it. Given that definition, then, “as well sleeping, as waking,” any observer doesn’t sense the table, but only one sensibilia.
Dream objects exist in the private space of the dreamer; they lose the correlation with other private spaces that real objects possess.
Such notions as “existence” and “non-existence” do not apply to sense-data, though they can apply to things described in terms of such data. The fact that a sense-datum exists, and that that datum is of a table, does not imply that a table exists. [Russell cites the fuller, symbol-based discussion of this point in Principia Mathematica, and also makes reference (p. 176) to the theory of descriptions.]
Russell goes on (pages 176-179) to indicate how his theoretical development answers four common arguments against realism:
(1) The same object can appear differently to different people. Russell allows different perspectives, which render these observational differences to be irrelevant for the question of object reality.
(2) The same object can give us seemingly incompatible sense-data. A stick in water can appear bent to the eye, though straight to the touch. But when a stick is a collection of sensibilia, and we recognize that the proper inference from one sensibile to another need not be fixed, incompatible sense-data are not a problem for a realist view of matter.
(3) Objects in dreams often are considered to be unreal—but the sense-data that lead to dreams are as real as any. As noted, dream sense-data lack the usual continuity and correlation with other sense-data. Nonetheless, they are physical, subject matter for physics to deal with.
(4) Hallucinations are like dreams, but it takes multiple observers for it to become clear that one person is suffering from hallucinations. (Dreams would have the same property if they were more closely connected to a person’s sense-data when awake.) The person suffering from hallucinations is in no position to know whether this is the case, or whether other people are conspiring against him.
“From the above instances it would appear that abnormal sense-data, of the kind which we regard as deceptive, have intrinsically just the same status as any others, but differ as regards their correlations or causal connections with other ‘sensibilia’ and with ‘things’ [p. 179].” The problem is not unreal data but rather, our unjustified expectations. As a result, abnormal sense-data do not present an impregnable obstacle for the science of physics. The approach provided above, though quite preliminary, especially as regards its role for time, permits physics to be empirically verifiable.
Wednesday, July 24, 2019
Mysticism and Logic, Chapter VIIIa
Chapter VIIIa, “The Relation of Sense-Data to Physics,” pages 145-164
Russell breaks down Chapter VIII into twelve sections; the chapter is sufficiently involved that my summentary itself will be broken into two posts. This first post (part “a” of Chapter VIII) covers the following sections:
I. The Problem Stated (p. 145)
II. Characteristics of Sense-Data (p. 147)
III. Sensibilia (p. 148)
IV. Sense-Data are Physical (p. 150)
V. ‘Sensibilia’ and ‘Things’ (p. 152)
VI. Constructions versus Inferences (p. 155)
VII. Private Space and the Space of Perspectives (p. 158)
VIII. The Placing of ‘Things’ and ‘Sensibilia’ in Perspective Space (p. 162).
The follow-up post, part b of Chapter VIII, will cover the remaining sections:
IX. The Definition of Matter (p. 164)
X. Time (p. 167)
XI. The Persistence of Things and Matter (p. 169)
XII. Illusions, Hallucinations, and Dreams (p. 173).
Now equipped with that barebones (yet formidable) context, on to the summentary of Chapter VIII…
Physics employs the usual scientific method of experiment and observation. But what we ultimately observe is limited by our senses, and that sense data is not the atoms and molecules themselves. What we think we know about atoms is through suspected correlations with the sense data. But how could such correlations be verified, given that only one side of the correlation, the sense data, will ever be known to us?
We could try to solve the inference problem by postulating some a priori truths: this is the route that philosophy often takes. The postulate-a-truth solution goes beyond experiment and observation, of course, which makes it inadvisable. Alternatively, we could define objects like atoms “as functions of sense-data [p. 146].”
The way we talk about physics is somewhat backwards. We say that when a certain type of wave meets our eyes, that certain colors are perceived. “But the waves are in fact inferred from the colours, not vice versa [p. 146].” So physics goes beyond experimental evidence to the extent that the waves are not themselves defined as functions of the data. From “stuff implies data” we need to move to “data imply stuff.”
We receive multiple sense data at any point, so it isn’t obvious of what a single sense datum consists. For our purposes, we can even accept a complex fact (A is to the left of B), as a sort of sense datum, even though, as opposed to a proper sense datum, the complex fact could be false.
Sense data exist when they are data, but whether the stuff that is sense data persist before or after when they are data is uncertain. “Sense-data at the times when they are data are all that we directly and primitively know of the external world [p. 148].” But there can be more than we know. [We are sort of like flatlanders trying to grasp 3D objects – RBR.] Physics (like metaphysics) in some sense deals with all the particulars, known to us or not. But the physics that we know of needs must deal only with sense data.
“Sensibilia” are the stuff akin to sense data, but without being sensed by any mind. (“Sensibile” is the singular form.) Sensibilia become sense-data by entering into a relationship (of acquaintance with a mind), like men become husbands by entering into a marital relationship. Can we infer (directly unobserved) sensibilia from sense data?
Sense data form “part of the actual subject matter of physics [p. 149].” Even when they are unobserved sensibilia, they are subject matter: observing sensibilia (and hence making them sense-data) adds only awareness to that which is already present.
“Physics” is related to “physical,” and Russell takes “physical” to refer to the stuff that is the subject matter for physics [!?]. A particular is “mental” if it is itself aware of something; facts are “mental” if they involve mental particulars. Russell hopes to show that sense data are physical – they might also be mental, but that is neither here or there for present purposes. [Russell (p. 151) notes that he does not accept the “new realist” position of Mach and James, though Russell’s discussion here is consistent with that position.] Sometimes the questions of the persistence and the physicality of sense data are conflated. Russell will argue that the data are physical – and hence within the scope of physics – though they probably do not persist in an unchanged way.
“Logically a sense-datum is an object, a particular of which the subject is aware [p. 152]” – and the subject is not a part of the sense-datum. The existence and the persistence of sense-data (or proto sense-data) do not require, of necessity, a sensing subject. The subject has sensations, his or her awareness of sense-data, and sensations are mental objects – though sense-data are physical.
We know that a table or other sensibilia appear differently to different people. But can a table (or other sensibile) in the same place simultaneously be both brown (to one observer) and yellow (to another)? Russell cites an article (pdf here) by T. P. Nunn for explaining how this subjectivity does not render sensibilia to be non-physical. Nunn’s solution notes that there are two “places” in question, the place at which the table appears and the place from which the table appears. Each observer’s place at which the table appears is not comparable to that of any other observer – though there can be correlations between these separate spaces. “No place in the private world of one observer is identical with a place in the private world of another observer [p. 154].” A table, then, could be the class of all appearances, or potential appearances, sensibilia, of the object in question. Though the appearances are not identical and cannot exist in the same place at the same time, the table is no less a physical concept – and we don’t need to adopt some ideal realm that contains the actual table.
Mathematical logic has developed the method of replacing a sort of imagined or inferred concept (like irrational numbers) with a constructed concept. Dr. Whitehead is the pioneer, and he suggested the application to physics of this approach to me [Russell].
“A complete application of the method which substitutes constructions for inferences would exhibit matter wholly in terms of sense-data, and even, we may add, of the sense-data of a single person, since the sense-data of others cannot be known without some element of inference [p. 157].” But we are far from achieving this ideal. In the meantime, we can discipline those inferences which cannot be avoided: they should be general, explicit, and similar to that stuff whose existence is already given – on this last principle, Kant’s thing-in-itself fails.
Russell permits two inferences: the sense-data of other observers (which uses analogy to accept the existence of other minds, and which rules out building a solipsistic basis for physics); and, the sensibilia that lack a current observer.
No sensibile can be a sense-datum to two observers simultaneously – though their sense-data will be similar, and two people can speak meaningfully of the same table. Everyone has their own private world of sense-data, different from everyone else’s. The place at which a sense-datum exists is a private space. There is no issue, then, with an object having two appearances in the same place, as those appearances exist in separate, observer-specific spaces. Multiple appearances of an object are not an argument against the physicality of the object.
“In addition to the private spaces belonging to the private worlds of different percipients, there is, however, another space, in which one whole private world counts as a point, or at least as a spatial unit [p. 159].” This is the space of perspectives, and its points (individual perspectives) do not require an actual observer to be present making perceptions. Nearby perspectives contain closely correlated sensibilia, and these sensibilia correspond to appearances of one object. Indeed, the object itself can be defined as the class of its appearances.
We can order all the perspectives of a thing in a space by taking similar views – those in which a penny looks perfectly circular, for instance – and arranging them by apparent size. The spatial order we end up with would have been replicated with any object that possessed the same set of appearances (though we could use an ordering metric other than size). “It is this empirical fact which has made it possible to construct the one all-embracing space of physics [p. 161].”
We now have a six-dimensional world: a three-dimensional collection of perspectives, where each perspective is itself three-dimensional. An object has associated with it many lines of perspective, and where they meet is itself a perspective, the one where the object is, the place “at which” it appears. But each perspective also provides its own place “from which” the object appears. Psychology is interested in studying sensibilia in the “from which” place, and physics is interested in studying sensibilia in the “at which” place.
Observers can order the appearances of an object by their proximity (to the mind of the observer, say); “those are nearer which belong to perspectives that are nearer to ‘the place where the thing is [p. 163].’” The fact that, by squinting, the appearance of an object changes, when we tend to suspect that the object itself does not change, is no longer a problem for regarding objects as physical. A thing is a class of appearances. If some appearances change – by squinting, say – then there is some change in the object. But we can define change in an object as occurring only when appearances that become arbitrarily close to the object also change. Squinting results in a change in something, but not in the object perceived.
Russell breaks down Chapter VIII into twelve sections; the chapter is sufficiently involved that my summentary itself will be broken into two posts. This first post (part “a” of Chapter VIII) covers the following sections:
I. The Problem Stated (p. 145)
II. Characteristics of Sense-Data (p. 147)
III. Sensibilia (p. 148)
IV. Sense-Data are Physical (p. 150)
V. ‘Sensibilia’ and ‘Things’ (p. 152)
VI. Constructions versus Inferences (p. 155)
VII. Private Space and the Space of Perspectives (p. 158)
VIII. The Placing of ‘Things’ and ‘Sensibilia’ in Perspective Space (p. 162).
The follow-up post, part b of Chapter VIII, will cover the remaining sections:
IX. The Definition of Matter (p. 164)
X. Time (p. 167)
XI. The Persistence of Things and Matter (p. 169)
XII. Illusions, Hallucinations, and Dreams (p. 173).
Now equipped with that barebones (yet formidable) context, on to the summentary of Chapter VIII…
Physics employs the usual scientific method of experiment and observation. But what we ultimately observe is limited by our senses, and that sense data is not the atoms and molecules themselves. What we think we know about atoms is through suspected correlations with the sense data. But how could such correlations be verified, given that only one side of the correlation, the sense data, will ever be known to us?
We could try to solve the inference problem by postulating some a priori truths: this is the route that philosophy often takes. The postulate-a-truth solution goes beyond experiment and observation, of course, which makes it inadvisable. Alternatively, we could define objects like atoms “as functions of sense-data [p. 146].”
The way we talk about physics is somewhat backwards. We say that when a certain type of wave meets our eyes, that certain colors are perceived. “But the waves are in fact inferred from the colours, not vice versa [p. 146].” So physics goes beyond experimental evidence to the extent that the waves are not themselves defined as functions of the data. From “stuff implies data” we need to move to “data imply stuff.”
We receive multiple sense data at any point, so it isn’t obvious of what a single sense datum consists. For our purposes, we can even accept a complex fact (A is to the left of B), as a sort of sense datum, even though, as opposed to a proper sense datum, the complex fact could be false.
Sense data exist when they are data, but whether the stuff that is sense data persist before or after when they are data is uncertain. “Sense-data at the times when they are data are all that we directly and primitively know of the external world [p. 148].” But there can be more than we know. [We are sort of like flatlanders trying to grasp 3D objects – RBR.] Physics (like metaphysics) in some sense deals with all the particulars, known to us or not. But the physics that we know of needs must deal only with sense data.
“Sensibilia” are the stuff akin to sense data, but without being sensed by any mind. (“Sensibile” is the singular form.) Sensibilia become sense-data by entering into a relationship (of acquaintance with a mind), like men become husbands by entering into a marital relationship. Can we infer (directly unobserved) sensibilia from sense data?
Sense data form “part of the actual subject matter of physics [p. 149].” Even when they are unobserved sensibilia, they are subject matter: observing sensibilia (and hence making them sense-data) adds only awareness to that which is already present.
“Physics” is related to “physical,” and Russell takes “physical” to refer to the stuff that is the subject matter for physics [!?]. A particular is “mental” if it is itself aware of something; facts are “mental” if they involve mental particulars. Russell hopes to show that sense data are physical – they might also be mental, but that is neither here or there for present purposes. [Russell (p. 151) notes that he does not accept the “new realist” position of Mach and James, though Russell’s discussion here is consistent with that position.] Sometimes the questions of the persistence and the physicality of sense data are conflated. Russell will argue that the data are physical – and hence within the scope of physics – though they probably do not persist in an unchanged way.
“Logically a sense-datum is an object, a particular of which the subject is aware [p. 152]” – and the subject is not a part of the sense-datum. The existence and the persistence of sense-data (or proto sense-data) do not require, of necessity, a sensing subject. The subject has sensations, his or her awareness of sense-data, and sensations are mental objects – though sense-data are physical.
We know that a table or other sensibilia appear differently to different people. But can a table (or other sensibile) in the same place simultaneously be both brown (to one observer) and yellow (to another)? Russell cites an article (pdf here) by T. P. Nunn for explaining how this subjectivity does not render sensibilia to be non-physical. Nunn’s solution notes that there are two “places” in question, the place at which the table appears and the place from which the table appears. Each observer’s place at which the table appears is not comparable to that of any other observer – though there can be correlations between these separate spaces. “No place in the private world of one observer is identical with a place in the private world of another observer [p. 154].” A table, then, could be the class of all appearances, or potential appearances, sensibilia, of the object in question. Though the appearances are not identical and cannot exist in the same place at the same time, the table is no less a physical concept – and we don’t need to adopt some ideal realm that contains the actual table.
Mathematical logic has developed the method of replacing a sort of imagined or inferred concept (like irrational numbers) with a constructed concept. Dr. Whitehead is the pioneer, and he suggested the application to physics of this approach to me [Russell].
“A complete application of the method which substitutes constructions for inferences would exhibit matter wholly in terms of sense-data, and even, we may add, of the sense-data of a single person, since the sense-data of others cannot be known without some element of inference [p. 157].” But we are far from achieving this ideal. In the meantime, we can discipline those inferences which cannot be avoided: they should be general, explicit, and similar to that stuff whose existence is already given – on this last principle, Kant’s thing-in-itself fails.
Russell permits two inferences: the sense-data of other observers (which uses analogy to accept the existence of other minds, and which rules out building a solipsistic basis for physics); and, the sensibilia that lack a current observer.
No sensibile can be a sense-datum to two observers simultaneously – though their sense-data will be similar, and two people can speak meaningfully of the same table. Everyone has their own private world of sense-data, different from everyone else’s. The place at which a sense-datum exists is a private space. There is no issue, then, with an object having two appearances in the same place, as those appearances exist in separate, observer-specific spaces. Multiple appearances of an object are not an argument against the physicality of the object.
“In addition to the private spaces belonging to the private worlds of different percipients, there is, however, another space, in which one whole private world counts as a point, or at least as a spatial unit [p. 159].” This is the space of perspectives, and its points (individual perspectives) do not require an actual observer to be present making perceptions. Nearby perspectives contain closely correlated sensibilia, and these sensibilia correspond to appearances of one object. Indeed, the object itself can be defined as the class of its appearances.
We can order all the perspectives of a thing in a space by taking similar views – those in which a penny looks perfectly circular, for instance – and arranging them by apparent size. The spatial order we end up with would have been replicated with any object that possessed the same set of appearances (though we could use an ordering metric other than size). “It is this empirical fact which has made it possible to construct the one all-embracing space of physics [p. 161].”
We now have a six-dimensional world: a three-dimensional collection of perspectives, where each perspective is itself three-dimensional. An object has associated with it many lines of perspective, and where they meet is itself a perspective, the one where the object is, the place “at which” it appears. But each perspective also provides its own place “from which” the object appears. Psychology is interested in studying sensibilia in the “from which” place, and physics is interested in studying sensibilia in the “at which” place.
Observers can order the appearances of an object by their proximity (to the mind of the observer, say); “those are nearer which belong to perspectives that are nearer to ‘the place where the thing is [p. 163].’” The fact that, by squinting, the appearance of an object changes, when we tend to suspect that the object itself does not change, is no longer a problem for regarding objects as physical. A thing is a class of appearances. If some appearances change – by squinting, say – then there is some change in the object. But we can define change in an object as occurring only when appearances that become arbitrarily close to the object also change. Squinting results in a change in something, but not in the object perceived.
Sunday, July 14, 2019
Mysticism and Logic, Chapter VII
“The Ultimate Constituents of Matter,” pages 125-144
The question “what is matter?” contains an insoluble part and a soluble part, and further, we know how to find the answer to the soluble portion.
The notion that mind and matter are distinct has not been popular with philosophers since the time of Leibniz. Matter itself has been problematized by physicists, and now it looks like a sort of electromagnetic field instead of chunks of palpable stuff. Further, we have learned that the senses through which we encounter matter don’t provide a single, true reckoning: “all our senses are liable to be affected by anything which affects the brain, like alcohol or hasheesh [p. 126].” We cannot trust our senses, including common sense, on the nature of matter.
The commonsensical notions that the stuff we perceive is physical; that it exists outside of our own mind; and, that it persists even when we avert our gaze, are flawed. In particular, the belief in object persistence needs to be re-examined. Following Bergson, objects are like characters in a film: what we perceive in a film to be a persistent person fleeing the police is actually a series of momentary people in close proximity. [Russell says (p. 128) that he heard Bergson’s analogy before he (Russell) had ever seen a movie (“cinematograph”), and that his first trip to the cinema was motivated by a desire to test Bergson’s claim.] So it is with actual men, and tables, and stars. Notice how this view extends to time what we already think about space: an object that “fills” a cubic foot actually consists of many smaller objects in close proximity.
The stuff being arranged in space or time is called [by Russell] “particulars.” The relations among particulars produce the patterns we perceive, the macro objects, which are “logical constructions [p. 129].” Particulars are like individual notes in a symphony – they are ephemeral, but we take notice of their relations. A table is similar, not to a trombone, but to the role of a trombone in a symphony – and is equally amorphous.
When I see a lightning flash, though I experience the flash mentally, through the sense of vision, the flash would still exist in the same way if I were unchanged except that I had lost my mental capacity, and hence could no longer sense the flash. The thing I see is separate from my sight. But people might not accept this claim, in particular, by suggesting that objects (or attributes, like color) cannot really exist outside the mind that notes them.
What does it mean to say that something is “in” the mind? It doesn’t mean that the “thing” is there in a spatial manner (though perhaps “in the brain” does mean that, but physically existing in the brain is not what people intend when they say that a quality like color exists in the mind). Colors are not like beliefs, which seem to be mental, without an external physical manifestation. Though a fire can make us experience pain, the fire itself need not be mental, just because the experience is mediated through our senses and mind.
Those who hold that objects like tables are mental, that they depend upon the observer, mistake the body for the mind. Yes, my perception of an object changes if I squint (or use eyeglasses), but the changes “are to be explained by physiology and optics, not by psychology [p. 134].” The visual representation that I have with eyeglasses disappears when I remove the glasses, but that does not provide evidence that the object itself vanishes. Our visual representations are not “ultimate constituents of matter,” but the whole argument is rendered moot if those ultimate constituents are Bergson-esque, restricted in space and time.
Physics tells us that what we call the sun is 93 million miles away, and that the electromagnetic waves that reach our eyes were emitted from that distance some eight minutes prior to reaching us. But our experience of the sun starts not with the release of the waves, but with the last step, our brain’s coding of the information from the eyes and optic nerve.
Events have the potential to have many different causes, not a single “cause.” One set of causes of “seeing the sun” involves the eyes, nerves, and brain. But we could list other antecedents, not involving these body parts, that possess an equal causal claim. In the case of seeing the sun, we could consider the sun and our eyes and brain as “assemblages of momentary particulars [p. 137]” – that is, the matter that we often take as real is itself a “logical construction,” and the sense data of an observer is the set of particulars caught in the observer’s snapshot or film, as modified by other particulars (such as those corresponding to the observer’s brain). The universe is thus a multiplex: “there are all those [three-dimensional spaces] perceived by observers, and presumably also those which are not perceived, merely because no observer is suitably situated for perceiving them [p. 139].”
This view leads to a six-dimensional space of particulars: the space of a set of particulars (a table, say) is itself three-dimensional, and the positions among sets of particulars can be specified with a further three dimensions.
An observer has a perspective, all that the observer observes, and objects are a correlated set of particulars. One can (often) classify particulars from the perspective viewpoint or the object viewpoint. (Some particulars, like dreams, might not be subject to these dual viewpoints.) Nevertheless, “[w]e cannot define a perspective as all the data of one percipient at one time, because we wish to allow the possibility of perspectives which are not perceived by any one [p. 140].”
Cue time – in “particular,” observer-specific time. The perspective associated with a particular is all the particulars simultaneous with that particular (p. 141). The timeline associated with a particular is its “biography.” As particulars need not be perceived, biographies need not be lived. [Russell calls unlived biographies “official,” in the sense, I believe, of committee membership being ex officio.]
Consider a particular with respect to one perspective. Shift the perspective marginally, and you will get a very similar particular – and this similarity is independent of the rest of the universe. When we think of a specific “thing,” it is the continuity with neighboring perspectives, and the independence from all else, which gives us the class of particulars constituting that “thing.” Physicists generally focus on things when they examine particulars, whereas psychologists focus on the perspective and “biography” associated with one observer.
The view propounded here (concerning the connection between sense-data and the physical world) is not intended to shed light on physics – but it is intended to shed light on standard psychological or metaphysical claims of the mental underpinnings of sense-data. Those standard claims often unduly favor permanence in the constituents of things, and draw on confused views about space and sense-data. Mind is not necessary for the existence of sense data, and “sense-data are merely those among the ultimate constituents of the physical world, of which we happen to be immediately aware; they themselves are purely physical, and all that is mental in connection with them is our awareness of them, which is irrelevant to their nature and to their place in physics [p. 143].”
The theory presented here suggests that there is no conflict between physics and psychology. The ultimate constituents of matter, such that physical things are a series of classes of particulars, means that physics can classify the particulars in one way when discussing matter, and psychologists can classify the particulars in another way (yielding perspectives and biographies).
Is the theory true? It could be, which is more than can be said for most alternative approaches to the question of matter. Further, the theory suggests a starting point from which a tolerable solution eventually can be devised.
The question “what is matter?” contains an insoluble part and a soluble part, and further, we know how to find the answer to the soluble portion.
The notion that mind and matter are distinct has not been popular with philosophers since the time of Leibniz. Matter itself has been problematized by physicists, and now it looks like a sort of electromagnetic field instead of chunks of palpable stuff. Further, we have learned that the senses through which we encounter matter don’t provide a single, true reckoning: “all our senses are liable to be affected by anything which affects the brain, like alcohol or hasheesh [p. 126].” We cannot trust our senses, including common sense, on the nature of matter.
The commonsensical notions that the stuff we perceive is physical; that it exists outside of our own mind; and, that it persists even when we avert our gaze, are flawed. In particular, the belief in object persistence needs to be re-examined. Following Bergson, objects are like characters in a film: what we perceive in a film to be a persistent person fleeing the police is actually a series of momentary people in close proximity. [Russell says (p. 128) that he heard Bergson’s analogy before he (Russell) had ever seen a movie (“cinematograph”), and that his first trip to the cinema was motivated by a desire to test Bergson’s claim.] So it is with actual men, and tables, and stars. Notice how this view extends to time what we already think about space: an object that “fills” a cubic foot actually consists of many smaller objects in close proximity.
The stuff being arranged in space or time is called [by Russell] “particulars.” The relations among particulars produce the patterns we perceive, the macro objects, which are “logical constructions [p. 129].” Particulars are like individual notes in a symphony – they are ephemeral, but we take notice of their relations. A table is similar, not to a trombone, but to the role of a trombone in a symphony – and is equally amorphous.
When I see a lightning flash, though I experience the flash mentally, through the sense of vision, the flash would still exist in the same way if I were unchanged except that I had lost my mental capacity, and hence could no longer sense the flash. The thing I see is separate from my sight. But people might not accept this claim, in particular, by suggesting that objects (or attributes, like color) cannot really exist outside the mind that notes them.
What does it mean to say that something is “in” the mind? It doesn’t mean that the “thing” is there in a spatial manner (though perhaps “in the brain” does mean that, but physically existing in the brain is not what people intend when they say that a quality like color exists in the mind). Colors are not like beliefs, which seem to be mental, without an external physical manifestation. Though a fire can make us experience pain, the fire itself need not be mental, just because the experience is mediated through our senses and mind.
Those who hold that objects like tables are mental, that they depend upon the observer, mistake the body for the mind. Yes, my perception of an object changes if I squint (or use eyeglasses), but the changes “are to be explained by physiology and optics, not by psychology [p. 134].” The visual representation that I have with eyeglasses disappears when I remove the glasses, but that does not provide evidence that the object itself vanishes. Our visual representations are not “ultimate constituents of matter,” but the whole argument is rendered moot if those ultimate constituents are Bergson-esque, restricted in space and time.
Physics tells us that what we call the sun is 93 million miles away, and that the electromagnetic waves that reach our eyes were emitted from that distance some eight minutes prior to reaching us. But our experience of the sun starts not with the release of the waves, but with the last step, our brain’s coding of the information from the eyes and optic nerve.
Events have the potential to have many different causes, not a single “cause.” One set of causes of “seeing the sun” involves the eyes, nerves, and brain. But we could list other antecedents, not involving these body parts, that possess an equal causal claim. In the case of seeing the sun, we could consider the sun and our eyes and brain as “assemblages of momentary particulars [p. 137]” – that is, the matter that we often take as real is itself a “logical construction,” and the sense data of an observer is the set of particulars caught in the observer’s snapshot or film, as modified by other particulars (such as those corresponding to the observer’s brain). The universe is thus a multiplex: “there are all those [three-dimensional spaces] perceived by observers, and presumably also those which are not perceived, merely because no observer is suitably situated for perceiving them [p. 139].”
This view leads to a six-dimensional space of particulars: the space of a set of particulars (a table, say) is itself three-dimensional, and the positions among sets of particulars can be specified with a further three dimensions.
An observer has a perspective, all that the observer observes, and objects are a correlated set of particulars. One can (often) classify particulars from the perspective viewpoint or the object viewpoint. (Some particulars, like dreams, might not be subject to these dual viewpoints.) Nevertheless, “[w]e cannot define a perspective as all the data of one percipient at one time, because we wish to allow the possibility of perspectives which are not perceived by any one [p. 140].”
Cue time – in “particular,” observer-specific time. The perspective associated with a particular is all the particulars simultaneous with that particular (p. 141). The timeline associated with a particular is its “biography.” As particulars need not be perceived, biographies need not be lived. [Russell calls unlived biographies “official,” in the sense, I believe, of committee membership being ex officio.]
Consider a particular with respect to one perspective. Shift the perspective marginally, and you will get a very similar particular – and this similarity is independent of the rest of the universe. When we think of a specific “thing,” it is the continuity with neighboring perspectives, and the independence from all else, which gives us the class of particulars constituting that “thing.” Physicists generally focus on things when they examine particulars, whereas psychologists focus on the perspective and “biography” associated with one observer.
The view propounded here (concerning the connection between sense-data and the physical world) is not intended to shed light on physics – but it is intended to shed light on standard psychological or metaphysical claims of the mental underpinnings of sense-data. Those standard claims often unduly favor permanence in the constituents of things, and draw on confused views about space and sense-data. Mind is not necessary for the existence of sense data, and “sense-data are merely those among the ultimate constituents of the physical world, of which we happen to be immediately aware; they themselves are purely physical, and all that is mental in connection with them is our awareness of them, which is irrelevant to their nature and to their place in physics [p. 143].”
The theory presented here suggests that there is no conflict between physics and psychology. The ultimate constituents of matter, such that physical things are a series of classes of particulars, means that physics can classify the particulars in one way when discussing matter, and psychologists can classify the particulars in another way (yielding perspectives and biographies).
Is the theory true? It could be, which is more than can be said for most alternative approaches to the question of matter. Further, the theory suggests a starting point from which a tolerable solution eventually can be devised.
Wednesday, June 26, 2019
Mysticism and Logic, Halftime
Time to take a halftime break in Mysticism and Logic. What have we learned in the first half?
Both intuition and logic are part of the human condition, akin to Plato’s two horses or Kahneman's System 1 and System 2. Intuition is particularly well suited to guiding quick judgments in dangerous circumstances; these circumstances, however, are not as common as they were when the intuitive decision capability evolved. As a result, intuition is often a source of error, probably, on net, a disadvantage for philosophy. What intuition or mystical insights can do is to suggest hypotheses and incentivize us to examine scientific or philosophical questions – these are the benefits of intuition in increasing our understanding. But once the incentivizing is done, the analysis must be conducted in a disinterested fashion, unaffected by our hopes and fears. We must examine the world as it is, not as we might wish it to be. Mysticism can provide an attitude, a stance, that promotes understanding, but reason is the tool for uncovering truths. Intuitive reflection also can aid rational inquiry through its frequent championing of a sort of detachment, which is the proper attitude for scientific investigation.
Philosophy can and should employ the methods of science, but not try to base reasoning on specific scientific results (which themselves are only approximate truths). When philosophers draw on evolution, for instance, they also adopt the unscientific (but personally comforting) viewpoint that things trend upwards, that there is a built-in progressive slant. Progress in philosophy cannot be secured by imposing preconditions on how the world must evolve.
An inappropriate certitude, the belief that a flash of insight reveals an unerring truth, is a negative consequence of mysticism. The common, intuitive sense of a grand unity also undermines rationality, as the personal interest of philosophers becomes to force reality into their preconceptions of this underlying harmony. Structures of philosophical thought can then be impressive and beautiful, but they are fragile: nothing remains for future thinkers to build on, because the foundations are faulty. In other sciences, mistaken ideas can nevertheless prove fruitful over time, as the errors are corrected while stronger elements remain. As a result, science has made great gains over the centuries – but it is hard to discern progress in philosophy, where past contributions collapse wholesale upon their shaky premises.
The usual argument in favor of science – that it produces technological wonders that make our lives better – is secondary to its chief virtue, that the outlook encouraged by science improves our way of thinking. And for Russell, education essentially is such an outlook, one that can broaden (temporally and geographically) our mental realms, in service to our primary desires. (One mark of a poor education is that it seeks to counter those primary desires.) Education can allow us to overcome our parochial blinkers. Beware of static disciplines, those that are too much in thrall to the past, as they narrow thought.
Chapters IV, V, and VI of Mysticism and Logic are devoted to the disciplines of mathematics and philosophy. Mathematics seeks truth, and attains beauty. Students can glimpse this beauty not by memorizing rules or by dwelling on the long-term material advantages tied to mathematical understanding, but by starting with examples, and then travelling the most pristine paths that lead to the general truths underlying the examples. Mathematics encourages and rewards the devotion to truth.
Mathematics only recently has become rigorous, thanks to advances (impressive, amazing advances) with respect to infinities, infinitesimals, continuity, and symbolic logic. Indeed, mathematics now can be seen as an application of logic. Some philosophical issues, such as Zeno’s paradoxes, have been cleared up in the process, and Euclid is relegated to historical interest. The further application of the full armory of mathematical logic could unleash a new golden age in philosophy.
Russell adopts a sort of Marxian view of prevailing ethics, that they serve the desires or interests of a subset of mankind, not universal truths. Indeed, we should recognize the limitations of those scientific “universal” truths that we do know, which we cannot trust to hold beyond the small piece of the universe we have been able to examine. Philosophy needs the scientific method to progress, but one part of the scientific posture is the notion that truths are provisional.
If there were a “which chapter is unlike the rest?” quiz for the first half of Mysticism and Logic, “A Free Man’s Worship” would be the obvious answer. The message here starts with the unavoidable recognition that the uncaring universe is not built for progress, or for us, or for our happiness. Each of us is a brief candle, soon to be extinguished. The key is to seize upon this recognition with gusto, to nourish and cling to our ideals, and to strive to use our flame, while we can, to aid and illuminate, to engage in the noble task of rendering this insignificant corner of space and time a somewhat better, more beautiful dwelling.
And so we move on to the second half of Mysticism and Logic…
Both intuition and logic are part of the human condition, akin to Plato’s two horses or Kahneman's System 1 and System 2. Intuition is particularly well suited to guiding quick judgments in dangerous circumstances; these circumstances, however, are not as common as they were when the intuitive decision capability evolved. As a result, intuition is often a source of error, probably, on net, a disadvantage for philosophy. What intuition or mystical insights can do is to suggest hypotheses and incentivize us to examine scientific or philosophical questions – these are the benefits of intuition in increasing our understanding. But once the incentivizing is done, the analysis must be conducted in a disinterested fashion, unaffected by our hopes and fears. We must examine the world as it is, not as we might wish it to be. Mysticism can provide an attitude, a stance, that promotes understanding, but reason is the tool for uncovering truths. Intuitive reflection also can aid rational inquiry through its frequent championing of a sort of detachment, which is the proper attitude for scientific investigation.
Philosophy can and should employ the methods of science, but not try to base reasoning on specific scientific results (which themselves are only approximate truths). When philosophers draw on evolution, for instance, they also adopt the unscientific (but personally comforting) viewpoint that things trend upwards, that there is a built-in progressive slant. Progress in philosophy cannot be secured by imposing preconditions on how the world must evolve.
An inappropriate certitude, the belief that a flash of insight reveals an unerring truth, is a negative consequence of mysticism. The common, intuitive sense of a grand unity also undermines rationality, as the personal interest of philosophers becomes to force reality into their preconceptions of this underlying harmony. Structures of philosophical thought can then be impressive and beautiful, but they are fragile: nothing remains for future thinkers to build on, because the foundations are faulty. In other sciences, mistaken ideas can nevertheless prove fruitful over time, as the errors are corrected while stronger elements remain. As a result, science has made great gains over the centuries – but it is hard to discern progress in philosophy, where past contributions collapse wholesale upon their shaky premises.
The usual argument in favor of science – that it produces technological wonders that make our lives better – is secondary to its chief virtue, that the outlook encouraged by science improves our way of thinking. And for Russell, education essentially is such an outlook, one that can broaden (temporally and geographically) our mental realms, in service to our primary desires. (One mark of a poor education is that it seeks to counter those primary desires.) Education can allow us to overcome our parochial blinkers. Beware of static disciplines, those that are too much in thrall to the past, as they narrow thought.
Chapters IV, V, and VI of Mysticism and Logic are devoted to the disciplines of mathematics and philosophy. Mathematics seeks truth, and attains beauty. Students can glimpse this beauty not by memorizing rules or by dwelling on the long-term material advantages tied to mathematical understanding, but by starting with examples, and then travelling the most pristine paths that lead to the general truths underlying the examples. Mathematics encourages and rewards the devotion to truth.
Mathematics only recently has become rigorous, thanks to advances (impressive, amazing advances) with respect to infinities, infinitesimals, continuity, and symbolic logic. Indeed, mathematics now can be seen as an application of logic. Some philosophical issues, such as Zeno’s paradoxes, have been cleared up in the process, and Euclid is relegated to historical interest. The further application of the full armory of mathematical logic could unleash a new golden age in philosophy.
Russell adopts a sort of Marxian view of prevailing ethics, that they serve the desires or interests of a subset of mankind, not universal truths. Indeed, we should recognize the limitations of those scientific “universal” truths that we do know, which we cannot trust to hold beyond the small piece of the universe we have been able to examine. Philosophy needs the scientific method to progress, but one part of the scientific posture is the notion that truths are provisional.
If there were a “which chapter is unlike the rest?” quiz for the first half of Mysticism and Logic, “A Free Man’s Worship” would be the obvious answer. The message here starts with the unavoidable recognition that the uncaring universe is not built for progress, or for us, or for our happiness. Each of us is a brief candle, soon to be extinguished. The key is to seize upon this recognition with gusto, to nourish and cling to our ideals, and to strive to use our flame, while we can, to aid and illuminate, to engage in the noble task of rendering this insignificant corner of space and time a somewhat better, more beautiful dwelling.
And so we move on to the second half of Mysticism and Logic…
Saturday, September 22, 2018
Mysticism and Logic, Chapter VI
“On Scientific Method in Philosophy,” pages 97-124
People typically are led to philosophical contemplation either by religious/moral concerns (Plato and Hegel, say), or by scientific interests (Hume and Locke, for example) – or by a healthy mixture of these motives (e.g., Aristotle, Kant). “Herbert Spencer, in whose honour we are assembled to-day, would naturally be classed among scientific philosophers…[p. 97].” Nonetheless, religion and ethics are also central to Spencer’s thought, and to his attachment to the notion of evolution.
Despite the creativity spawned through ethical and religious motives, their net impact on philosophy has been negative. Science, at least the kind that is divorced from those same sorts of religious motives, should be the driving force in the future of philosophy. Within the realm of science, however, it is the methods of inquiry, and not the cutting-edge results, that are positioned to be profitably borrowed by philosophers.
The pull of religion and ethics on philosophy is evident in the amount of cogitation on the “universe” and on “good and evil.” All this talk of the universe is a leftover from a pre-Copernican mindset, where the key position of mankind derived from the notion that the earth itself is central to the cosmos. A related vestige is the quick resort to claims of a oneness to the universe. Post-Copernicus, we should recognize “that the apparent oneness of the world is merely the oneness of what is seen by a single spectator or apprehended by a single mind [p. 99].”
[Russell (page 100) initiates the first of two numbered but untitled sections -- largely via a lengthy quoted passage from William James about how the ability to conceive of different universes and to collect them under a common name does not imply any connection between them, any “oneness”.] Nevertheless, there are two smaller unities, one involving that subset of existence experienced by an individual consciousness, and a second concerning the constancy of scientific laws in the small piece of the cosmos known to us. But neither of these unities holds any external legitimacy, permits any valid generalizations beyond their sphere. General laws in physics are unavoidable, as there is a finite amount of particles and thus their complete data forms a sort of (degenerate) general law; “what is surprising in physics is not the existence of general laws, but their extreme simplicity [p. 102].” The rules are so simple that even we can discover them. But again, there’s no reason to expect that those rules remain simple outside of sample; indeed, the rules we have identified must perforce be non-complex, otherwise we would not have discovered them.
More generally, the data that we have collected are not necessarily representative of all that exists -- data are selectively encountered. And those general scientific results we possess are rather infirm, particularly likely to be overturned as we learn more. We must ensure that any philosophical insights that we deduce from scientific results are those that largely will withstand the likely future modifications to those results. We should be wary, for example, of being too dependent on the supposed conservation of energy or mass. Both mass and energy are proving to be more complex than previously thought, and while within the traditional realm of physical sciences the old results are sturdy, out-of-sample generalizations requiring the conservation of some measure of energy or mass are unfounded.
Philosophies developed around evolution, whether older (Hegel, Spencer) or more modern (Pragmatism, Bergson), involve a normative bias, a notion that evolution is progress. [Bergson and progress came up earlier in “Mysticism and Logic”.] The evolutionary philosophies (though not Hegel’s) tend to use biological evolution as their guide. But our biological sample is quite limited, and even within it, the claim of progress (“from the protozoon to the philosopher [p. 106]”) is made by those who think of themselves as, to date, the apex of this supposed upwards movement. More generally, the ethical ideas employed by those whose philosophical inquiries are morally motivated are human-centered, and constitute “an attempt, however veiled, to legislate for the universe on the basis of the present desires of men [p. 107].” Hopes get in the way of facts.
Ethics are really about social affiliation, offering justifications for the actions of the group to which one belongs. The fact that ethical schools support some (seemingly) socially desirable behaviors, such as self-sacrifice, does not undermine their ultimate, action-laundering purpose. The lack of neutrality in ethics is what renders moral considerations unsuitable as a complete basis for philosophical reasoning – even though there is much of practical value in some ethically-based philosophical approaches, such as that of Spinoza. A scientific philosophy must proceed upon facts, not upon hopes for human progress.
[Page 110 begins the second of Russell’s two numbered but untitled sections.] After spurious ideas about the universe and the good are expunged, we see that philosophical propositions must not depend on any specific worlds, but apply to all possible worlds: such are the sort of general propositions found in logic. While such propositions apply to all things separately, they say nothing about any universe, any collection of these separate things. “The philosophy which I wish to advocate may be called logical atomism or absolute pluralism, because, while maintaining that there are many things, it denies that there is a whole composed of those things [p. 111].” Philosophical claims must be a priori, incapable of being proven or disproven through empirical data: arguments built around the path of history, for instance, are not of this nature. In brief, “philosophy is the science of the possible [p. 111, italics BR’s],” or the general.
By this reckoning, philosophy and logic are identical. Logic is general: note the use of variables in stating propositions; further, logic identifies the forms of propositions that can apply to these general facts, it yields “an inventory of possibilities [p. 112].” Perhaps surprisingly, specific areas of inquiry, such as those concerning space and time, are hindered by a lack of understanding of logical forms; that is, particular sciences can still be helped by developments in logic.
Science has made definite progress over the centuries, but the same cannot be said for philosophy. Every philosopher starts out afresh, with new fundamentals, and then the shortcomings in those fundamentals render the entire chain of reasoning to be erroneous: there are no partial truths in this style of philosophy, nothing that can serve as a starting point for later advances. “A scientific philosophy such as I wish to recommend will be piecemeal and tentative like other sciences; above all, it will be able to invent hypotheses which, even if they are not wholly true, will yet remain fruitful after the necessary corrections have been made [p. 113].” Like science, philosophy can then make better and better approximations to the truth. Philosophers should not dream up grand systems; rather, they should look to break current conundrums into smaller questions that individually are susceptible, with the correct logical forms, to solution.
Consider the question of space as promulgated by Kant’s Transcendental Aesthetic [good explanatory lecture pdf here]. Kant’s problem is comprised of three separate questions in different areas of inquiry: logic, physics, and the theory of knowledge. The theory of knowledge issue is the thorny one, the one we are furthest from solving.
The logical problem involves recognizing that what is key about the geometry of space is not so much the underlying axioms that serve as the foundation of a specific geometry, but rather how points in the space can be (partially) ordered by a “betweenness” criterion, where it can be ascertained if point B lies between points A and C. There are many geometries that share the same betweenness criterion, and geometrical reasoning takes place in a strictly logical fashion at this more general level of relations, as opposed to acting on some underlying, and less general, set of axioms.
The physical problem is how we connect real-world objects to mathematical entities such as points and planes – after all, mathematical physics has proven itself to teach us quite a bit about actual objects, even though those objects do not meet the definitions employed in the mathematics. A. N. Whitehead has shown the way, to understand a point, for instance, as the class of all physical objects that contain the point (p. 117). This approach does not require that we assume that objects are made of points, but helps explain how theories based on points nevertheless have real-world relevance.
Kant’s concern about how we can have a priori knowledge of geometry is softened (or eliminated) when we distinguish the logic of geometry from its physical manifestations. We can have a priori knowledge of the logic, but our physical knowledge is synthetic, and only approximates the logical constructs. Kant’s worry about how we can have synthetic, a priori knowledge of geometry is answered by recognizing that we lack such knowledge. We don’t know what happens when we look at actual parallel lines in space, so we shouldn’t act as if we have a priori knowledge of them: those who claim such knowledge have taken a very constrained view of the nature of space.
A similar style of analysis can help clarify the reality of what we perceive and if that reality is independent of the observer. The objects we perceive might be like the supposed noise in a forest, non-existent when not perceived – and there is no way we could tell if this is the case. Independence is as ineffable as reality: we can always identify multiple channels of causation of events, so that in the end, there is only correlation, not causation. “The view which I should wish to advocate is that objects of perception do not persist unchanged at times when they are not perceived, although probably objects more or less resembling them do exist at such times…[p. 123].” [Russell points to his 1914 book, Our Knowledge of the External World, for elaboration.]
So a scientific approach to philosophy requires that we farm out a subset of questions to allied sciences; that we accept that answers to some other questions are beyond human capabilities; and, that with the analytic method of being careful with definitions and decomposing large questions into bite-sized chunks, philosophers can make the slow and steady progress that marks science more generally.
People typically are led to philosophical contemplation either by religious/moral concerns (Plato and Hegel, say), or by scientific interests (Hume and Locke, for example) – or by a healthy mixture of these motives (e.g., Aristotle, Kant). “Herbert Spencer, in whose honour we are assembled to-day, would naturally be classed among scientific philosophers…[p. 97].” Nonetheless, religion and ethics are also central to Spencer’s thought, and to his attachment to the notion of evolution.
Despite the creativity spawned through ethical and religious motives, their net impact on philosophy has been negative. Science, at least the kind that is divorced from those same sorts of religious motives, should be the driving force in the future of philosophy. Within the realm of science, however, it is the methods of inquiry, and not the cutting-edge results, that are positioned to be profitably borrowed by philosophers.
The pull of religion and ethics on philosophy is evident in the amount of cogitation on the “universe” and on “good and evil.” All this talk of the universe is a leftover from a pre-Copernican mindset, where the key position of mankind derived from the notion that the earth itself is central to the cosmos. A related vestige is the quick resort to claims of a oneness to the universe. Post-Copernicus, we should recognize “that the apparent oneness of the world is merely the oneness of what is seen by a single spectator or apprehended by a single mind [p. 99].”
[Russell (page 100) initiates the first of two numbered but untitled sections -- largely via a lengthy quoted passage from William James about how the ability to conceive of different universes and to collect them under a common name does not imply any connection between them, any “oneness”.] Nevertheless, there are two smaller unities, one involving that subset of existence experienced by an individual consciousness, and a second concerning the constancy of scientific laws in the small piece of the cosmos known to us. But neither of these unities holds any external legitimacy, permits any valid generalizations beyond their sphere. General laws in physics are unavoidable, as there is a finite amount of particles and thus their complete data forms a sort of (degenerate) general law; “what is surprising in physics is not the existence of general laws, but their extreme simplicity [p. 102].” The rules are so simple that even we can discover them. But again, there’s no reason to expect that those rules remain simple outside of sample; indeed, the rules we have identified must perforce be non-complex, otherwise we would not have discovered them.
More generally, the data that we have collected are not necessarily representative of all that exists -- data are selectively encountered. And those general scientific results we possess are rather infirm, particularly likely to be overturned as we learn more. We must ensure that any philosophical insights that we deduce from scientific results are those that largely will withstand the likely future modifications to those results. We should be wary, for example, of being too dependent on the supposed conservation of energy or mass. Both mass and energy are proving to be more complex than previously thought, and while within the traditional realm of physical sciences the old results are sturdy, out-of-sample generalizations requiring the conservation of some measure of energy or mass are unfounded.
Philosophies developed around evolution, whether older (Hegel, Spencer) or more modern (Pragmatism, Bergson), involve a normative bias, a notion that evolution is progress. [Bergson and progress came up earlier in “Mysticism and Logic”.] The evolutionary philosophies (though not Hegel’s) tend to use biological evolution as their guide. But our biological sample is quite limited, and even within it, the claim of progress (“from the protozoon to the philosopher [p. 106]”) is made by those who think of themselves as, to date, the apex of this supposed upwards movement. More generally, the ethical ideas employed by those whose philosophical inquiries are morally motivated are human-centered, and constitute “an attempt, however veiled, to legislate for the universe on the basis of the present desires of men [p. 107].” Hopes get in the way of facts.
Ethics are really about social affiliation, offering justifications for the actions of the group to which one belongs. The fact that ethical schools support some (seemingly) socially desirable behaviors, such as self-sacrifice, does not undermine their ultimate, action-laundering purpose. The lack of neutrality in ethics is what renders moral considerations unsuitable as a complete basis for philosophical reasoning – even though there is much of practical value in some ethically-based philosophical approaches, such as that of Spinoza. A scientific philosophy must proceed upon facts, not upon hopes for human progress.
[Page 110 begins the second of Russell’s two numbered but untitled sections.] After spurious ideas about the universe and the good are expunged, we see that philosophical propositions must not depend on any specific worlds, but apply to all possible worlds: such are the sort of general propositions found in logic. While such propositions apply to all things separately, they say nothing about any universe, any collection of these separate things. “The philosophy which I wish to advocate may be called logical atomism or absolute pluralism, because, while maintaining that there are many things, it denies that there is a whole composed of those things [p. 111].” Philosophical claims must be a priori, incapable of being proven or disproven through empirical data: arguments built around the path of history, for instance, are not of this nature. In brief, “philosophy is the science of the possible [p. 111, italics BR’s],” or the general.
By this reckoning, philosophy and logic are identical. Logic is general: note the use of variables in stating propositions; further, logic identifies the forms of propositions that can apply to these general facts, it yields “an inventory of possibilities [p. 112].” Perhaps surprisingly, specific areas of inquiry, such as those concerning space and time, are hindered by a lack of understanding of logical forms; that is, particular sciences can still be helped by developments in logic.
Science has made definite progress over the centuries, but the same cannot be said for philosophy. Every philosopher starts out afresh, with new fundamentals, and then the shortcomings in those fundamentals render the entire chain of reasoning to be erroneous: there are no partial truths in this style of philosophy, nothing that can serve as a starting point for later advances. “A scientific philosophy such as I wish to recommend will be piecemeal and tentative like other sciences; above all, it will be able to invent hypotheses which, even if they are not wholly true, will yet remain fruitful after the necessary corrections have been made [p. 113].” Like science, philosophy can then make better and better approximations to the truth. Philosophers should not dream up grand systems; rather, they should look to break current conundrums into smaller questions that individually are susceptible, with the correct logical forms, to solution.
Consider the question of space as promulgated by Kant’s Transcendental Aesthetic [good explanatory lecture pdf here]. Kant’s problem is comprised of three separate questions in different areas of inquiry: logic, physics, and the theory of knowledge. The theory of knowledge issue is the thorny one, the one we are furthest from solving.
The logical problem involves recognizing that what is key about the geometry of space is not so much the underlying axioms that serve as the foundation of a specific geometry, but rather how points in the space can be (partially) ordered by a “betweenness” criterion, where it can be ascertained if point B lies between points A and C. There are many geometries that share the same betweenness criterion, and geometrical reasoning takes place in a strictly logical fashion at this more general level of relations, as opposed to acting on some underlying, and less general, set of axioms.
The physical problem is how we connect real-world objects to mathematical entities such as points and planes – after all, mathematical physics has proven itself to teach us quite a bit about actual objects, even though those objects do not meet the definitions employed in the mathematics. A. N. Whitehead has shown the way, to understand a point, for instance, as the class of all physical objects that contain the point (p. 117). This approach does not require that we assume that objects are made of points, but helps explain how theories based on points nevertheless have real-world relevance.
Kant’s concern about how we can have a priori knowledge of geometry is softened (or eliminated) when we distinguish the logic of geometry from its physical manifestations. We can have a priori knowledge of the logic, but our physical knowledge is synthetic, and only approximates the logical constructs. Kant’s worry about how we can have synthetic, a priori knowledge of geometry is answered by recognizing that we lack such knowledge. We don’t know what happens when we look at actual parallel lines in space, so we shouldn’t act as if we have a priori knowledge of them: those who claim such knowledge have taken a very constrained view of the nature of space.
A similar style of analysis can help clarify the reality of what we perceive and if that reality is independent of the observer. The objects we perceive might be like the supposed noise in a forest, non-existent when not perceived – and there is no way we could tell if this is the case. Independence is as ineffable as reality: we can always identify multiple channels of causation of events, so that in the end, there is only correlation, not causation. “The view which I should wish to advocate is that objects of perception do not persist unchanged at times when they are not perceived, although probably objects more or less resembling them do exist at such times…[p. 123].” [Russell points to his 1914 book, Our Knowledge of the External World, for elaboration.]
So a scientific approach to philosophy requires that we farm out a subset of questions to allied sciences; that we accept that answers to some other questions are beyond human capabilities; and, that with the analytic method of being careful with definitions and decomposing large questions into bite-sized chunks, philosophers can make the slow and steady progress that marks science more generally.
Wednesday, August 29, 2018
Mysticism and Logic, Chapter V
“Mathematics and the Metaphysicians,” pages 74-96
[In the Preface (pages v-vi), Russell tells us that this chapter originally appeared in 1901, and that necessary updates are indicated in footnotes.]
Despite what you may have heard, pure mathematics is a recent discovery, made by Boole in 1854. Though unbeknownst even to Boole, mathematics and formal logic are equivalent. Pure mathematics is about general statements, along the lines of “If (some proposition) A is true, then (some proposition) B is true,” but precisely what A is and whether in fact it is true or not are issues for applied, not pure, mathematics. “Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true [p. 75].”
The method familiar from geometry – start with some primitive definitions and axioms, and then commence deducing – is not descriptive of pure mathematics, except as that method applies to unalloyed logic. The branches of mathematics (geometry, arithmetic, and so on) develop from the building blocks of logic. It used to be an item of contention in philosophy as to whether mathematics could be built solely upon logic; the mathematicians have ended the contention by actually doing the building.
Aristotle and his syllogism founded formal logic, which for centuries advanced no further. Since 1850, we advance more every decade than the cumulative progress “from Aristotle to Leibniz [p.76];” Charles Sanders Peirce’s Logic of Relatives has been instrumental in expanding the scope of logic.
Symbolic logic allows us to get at the foundations of mathematics, through the paradoxical method of making the initial part of the path more difficult. The symbolism eliminates any obviousness from even the simplest propositions (such as 2+2=4), so we must rely on mechanical operations. We can thereby uncover the minimal set of definitions and axioms to generate algebra, for instance. At first, it might seem frivolous to rigorously prove that 2+2=4, but by connecting obvious statements through the non-obvious applications of rigorous methods, we are learning. One thing that is learned is that obvious truths sometimes are false. For numbers in general, for instance, it is not the case that the addition of one item leads to a greater number of items (thanks to transfinite numbers).
Giuseppe Peano is at the forefront of mathematical logic. [Russell inserts a footnote indicating that in the original version of this chapter, he was unfamiliar with the work of Gottlob Frege, but that Frege should be included as a contemporary leader in logic.] Peano dispenses with words (including “therefore,” and “let us assume”) in developing most (and soon all) of mathematics to symbols. Excepting geometry, most mathematics needs only three primitives: zero, number, and “next after.” And even these three primitives can be replaced by two ideas, relation and class.
Leibniz glimpsed the method that Peano has developed, but Leibniz’s progress was constrained by his unwillingness to accept that Aristotle made logical errors. Though lampooned, Leibniz’s vision of philosophical disagreements resolved by calculations has, to a significant extent, been realized, at least in mathematical philosophy. What used to be mysteries (such as the nature of infinity) are now certainties.
For centuries it was believed that Aristotle had effectively answered the paradoxes of Zeno of Elea, but with the work of Karl Weierstrass, we learn that Zeno largely was right. Zeno’s sole mistake was to believe (if he did believe it) that the non-existence of a state of change implies an unchanging reality. Weierstrass’s use of mathematics avoids any mistaken inferences, with the result that Zeno’s paradoxes appear as straightforward statements, though perhaps at the cost of removing the delight that can accompany Zeno’s enigmas.
Zeno’s paradoxes fundamentally implicate “the problems of the infinitesimal, the infinite, and continuity [p. 81].” For centuries no serious progress was made on these problems, until Weierstrass, Dedekind, and Cantor solved them: “[t]his achievement is probably the greatest of which our age has to boast [p. 81].” Weierstrass, in particular, showed that the infinitesimal which had bedeviled thinkers for millennia could safely be dispensed with. We can always divide a length more finely without ever reaching a single point. We cannot say where a body in motion will be in the next instant, because there is no such thing as the next instant (p. 84).
Recent advances on infinity have taken rather the opposite path than new thinking on infinitesimals: in the case of infinity, a concept that once was thought to hold inherent logical contradictions has been rendered perfectly understandable by Dedekind and Cantor. Their first step was to offer a precise definition of infinity; Cantor then demonstrated that the standard contradictions disappear if a faulty line of reasoning employed in the proofs of the contradictions is rejected. That commonsensical but incorrect notion was that a proper subset of a collection has fewer items than the original collection. The notion is correct for a finite collection, but provides the definitional distinction for infinity: “A collection of terms is infinite when it contains as parts other collections which have just as many terms as it has [p. 86].” Since every positive integer has a unique double (which is even), the positive integers and the even integers have the same number of terms – and hence there is an infinite amount of positive integers.
In dealing with the infinite, we can’t determine the size of a set just by counting the terms – that process would never end. But of two infinite collections, we can ask (and determine) if one has more terms than the other. The method is like the even-number example: we look for a one-to-one mapping between the collections. If such a mapping exists, then the two infinite sets are of the same size. Some infinite collections are larger than others, however. Is there a greatest infinite number? Cantor says no, but his proof is mistaken. [Russell adds (page 89) a footnote in 1917 indicating that Cantor’s proof actually is correct, and that it was Russell who was mistaken.]
The paradoxcial Zenovian notion that fleet Achilles cannot catch a slow tortoise that possesses a head start goes away when we see that proper subsets do not have to be smaller than their parent collection. Why are people led to think that there is a serious argument that Achilles cannot catch the tortoise? Because people recognize that at each instant after the start of the race, Achilles must be in precisely one spot and the tortoise must be in precisely one spot. But with the head start, the tortoise necessarily has been in more spots along the race course than has Achilles. As each passing instant adds one new spot for Achilles and one more spot for the tortoise, Achilles can never have been in a greater number of spots than the tortoise: he cannot catch the tortoise. Once we recognize that Achilles’s portion of the course does not possess fewer spots than the tortoise’s longer portion of the course, however, the argument crumbles.
Russell offers “the paradox of Tristam Shandy [p.90],” building on Tristam’s recognition that as it takes him longer to write about a period of his life than the period itself, his autobiography will become increasingly further from completion even as he makes progress on it. But as we can match each day of Tristam’s life with the year that it takes to write about it — that one-to-one mapping again — then over infinite time, the autobiography is complete. For a similar reason, with enough time, the slow tortoise will go as far as fleet Achilles. With Cantor as a guide, the paradoxes that once seemed inherent to infinity no longer look so paradoxical -- just as scientific advances have rendered it uninteresting that people can live on the other side of the earth despite their ”necessity” to live upside down.
The puzzles of continuity likewise have given way to Cantor’s exactitude: continuity is one type of order. It is order, and not quantity, that now seems fundamental in mathematics; much can be accomplished without introducing numbers. Limits, formerly expressed as quantities, are now based upon order. “Thus, for instance, the smallest of the infinite integers is the limit of the finite integers, though all finite integers are at an infinite distance from it [p. 92].”
Geometry, too, appears in a different light as its non-Euclidean variants have multiplied. Geometry is not about the space in which we live: it is about the valid conclusions that can be drawn from some set of starting principles, where those principles need not accord with what we see in the real world. The leading books in geometry now do not even rely upon figures to demonstrate their arguments. Indeed, figures mislead, as they suggest conclusions that seem obvious from the figures, but do not follow with necessity from the first principles.
Modern geometry does not start with an assumption of some large space; rather, a point is assumed, and then a second distinct point, with lines and other points building upon these beginnings, so nothing is assumed to exist unless it is necessary for the next step of reasoning. The mathematicians responsible for much of this improvement are Peano and Fano. Euclid’s work itself now seems error-filled, in the sense that, strictly speaking, many of his theorems do not follow from his axioms alone. The difficulty of Euclid’s book, along with its errors, renders it unfit for any consideration beyond the historical; it should not be thrust upon English schoolboys. “A book should have either intelligibility or correctness; to combine the two is impossible, but to lack both is to be unworthy of such a place as Euclid has occupied in education [p. 95].” [Russell’s own youthful introduction to Euclid was both intelligible and revelatory. Recall that this chapter of Mysticism and Logic originally appeared in 1901; in 1902, Russell published a short essay with a more detailed critique of Euclid. Even earlier, Russell had dealt with the empirical validity of Euclid’s postulates, an issue he regards in Mysticism and Logic as “a comparatively trivial matter [p. 94].”]
Modern formalism and symbolic logic have brought rigor to mathematics, a rigor that was absent since the Ancient Greeks. Mathematical advances in the interim were so alluring that the foundations of the subject were unexamined. Weierstrass and his ilk were to mathematicians what Hume was to Kant, the prod that awoke a slumbering intellect. Formalism can seem pedantic, but its record in uncovering errors provides its justification.
Kant’s metaphysics cannot survive the fact that mathematics (including geometry) are but elements of symbolic logic: his theory of knowledge was meant to complete Euclid. Now that we know that Euclid is wrong, not simply incomplete, the Kantian theory is not viable. What is needed is for mathematical logic to come to full flower, and then for philosophy to repurpose the same rigorous tools. If this process is successful, the future might see a golden age in philosophy that parallels the recent era of advance in mathematics, and matches the glory that was philosophy in Ancient Greece.
[In the Preface (pages v-vi), Russell tells us that this chapter originally appeared in 1901, and that necessary updates are indicated in footnotes.]
Despite what you may have heard, pure mathematics is a recent discovery, made by Boole in 1854. Though unbeknownst even to Boole, mathematics and formal logic are equivalent. Pure mathematics is about general statements, along the lines of “If (some proposition) A is true, then (some proposition) B is true,” but precisely what A is and whether in fact it is true or not are issues for applied, not pure, mathematics. “Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true [p. 75].”
The method familiar from geometry – start with some primitive definitions and axioms, and then commence deducing – is not descriptive of pure mathematics, except as that method applies to unalloyed logic. The branches of mathematics (geometry, arithmetic, and so on) develop from the building blocks of logic. It used to be an item of contention in philosophy as to whether mathematics could be built solely upon logic; the mathematicians have ended the contention by actually doing the building.
Aristotle and his syllogism founded formal logic, which for centuries advanced no further. Since 1850, we advance more every decade than the cumulative progress “from Aristotle to Leibniz [p.76];” Charles Sanders Peirce’s Logic of Relatives has been instrumental in expanding the scope of logic.
Symbolic logic allows us to get at the foundations of mathematics, through the paradoxical method of making the initial part of the path more difficult. The symbolism eliminates any obviousness from even the simplest propositions (such as 2+2=4), so we must rely on mechanical operations. We can thereby uncover the minimal set of definitions and axioms to generate algebra, for instance. At first, it might seem frivolous to rigorously prove that 2+2=4, but by connecting obvious statements through the non-obvious applications of rigorous methods, we are learning. One thing that is learned is that obvious truths sometimes are false. For numbers in general, for instance, it is not the case that the addition of one item leads to a greater number of items (thanks to transfinite numbers).
Giuseppe Peano is at the forefront of mathematical logic. [Russell inserts a footnote indicating that in the original version of this chapter, he was unfamiliar with the work of Gottlob Frege, but that Frege should be included as a contemporary leader in logic.] Peano dispenses with words (including “therefore,” and “let us assume”) in developing most (and soon all) of mathematics to symbols. Excepting geometry, most mathematics needs only three primitives: zero, number, and “next after.” And even these three primitives can be replaced by two ideas, relation and class.
Leibniz glimpsed the method that Peano has developed, but Leibniz’s progress was constrained by his unwillingness to accept that Aristotle made logical errors. Though lampooned, Leibniz’s vision of philosophical disagreements resolved by calculations has, to a significant extent, been realized, at least in mathematical philosophy. What used to be mysteries (such as the nature of infinity) are now certainties.
For centuries it was believed that Aristotle had effectively answered the paradoxes of Zeno of Elea, but with the work of Karl Weierstrass, we learn that Zeno largely was right. Zeno’s sole mistake was to believe (if he did believe it) that the non-existence of a state of change implies an unchanging reality. Weierstrass’s use of mathematics avoids any mistaken inferences, with the result that Zeno’s paradoxes appear as straightforward statements, though perhaps at the cost of removing the delight that can accompany Zeno’s enigmas.
Zeno’s paradoxes fundamentally implicate “the problems of the infinitesimal, the infinite, and continuity [p. 81].” For centuries no serious progress was made on these problems, until Weierstrass, Dedekind, and Cantor solved them: “[t]his achievement is probably the greatest of which our age has to boast [p. 81].” Weierstrass, in particular, showed that the infinitesimal which had bedeviled thinkers for millennia could safely be dispensed with. We can always divide a length more finely without ever reaching a single point. We cannot say where a body in motion will be in the next instant, because there is no such thing as the next instant (p. 84).
Recent advances on infinity have taken rather the opposite path than new thinking on infinitesimals: in the case of infinity, a concept that once was thought to hold inherent logical contradictions has been rendered perfectly understandable by Dedekind and Cantor. Their first step was to offer a precise definition of infinity; Cantor then demonstrated that the standard contradictions disappear if a faulty line of reasoning employed in the proofs of the contradictions is rejected. That commonsensical but incorrect notion was that a proper subset of a collection has fewer items than the original collection. The notion is correct for a finite collection, but provides the definitional distinction for infinity: “A collection of terms is infinite when it contains as parts other collections which have just as many terms as it has [p. 86].” Since every positive integer has a unique double (which is even), the positive integers and the even integers have the same number of terms – and hence there is an infinite amount of positive integers.
In dealing with the infinite, we can’t determine the size of a set just by counting the terms – that process would never end. But of two infinite collections, we can ask (and determine) if one has more terms than the other. The method is like the even-number example: we look for a one-to-one mapping between the collections. If such a mapping exists, then the two infinite sets are of the same size. Some infinite collections are larger than others, however. Is there a greatest infinite number? Cantor says no, but his proof is mistaken. [Russell adds (page 89) a footnote in 1917 indicating that Cantor’s proof actually is correct, and that it was Russell who was mistaken.]
The paradoxcial Zenovian notion that fleet Achilles cannot catch a slow tortoise that possesses a head start goes away when we see that proper subsets do not have to be smaller than their parent collection. Why are people led to think that there is a serious argument that Achilles cannot catch the tortoise? Because people recognize that at each instant after the start of the race, Achilles must be in precisely one spot and the tortoise must be in precisely one spot. But with the head start, the tortoise necessarily has been in more spots along the race course than has Achilles. As each passing instant adds one new spot for Achilles and one more spot for the tortoise, Achilles can never have been in a greater number of spots than the tortoise: he cannot catch the tortoise. Once we recognize that Achilles’s portion of the course does not possess fewer spots than the tortoise’s longer portion of the course, however, the argument crumbles.
Russell offers “the paradox of Tristam Shandy [p.90],” building on Tristam’s recognition that as it takes him longer to write about a period of his life than the period itself, his autobiography will become increasingly further from completion even as he makes progress on it. But as we can match each day of Tristam’s life with the year that it takes to write about it — that one-to-one mapping again — then over infinite time, the autobiography is complete. For a similar reason, with enough time, the slow tortoise will go as far as fleet Achilles. With Cantor as a guide, the paradoxes that once seemed inherent to infinity no longer look so paradoxical -- just as scientific advances have rendered it uninteresting that people can live on the other side of the earth despite their ”necessity” to live upside down.
The puzzles of continuity likewise have given way to Cantor’s exactitude: continuity is one type of order. It is order, and not quantity, that now seems fundamental in mathematics; much can be accomplished without introducing numbers. Limits, formerly expressed as quantities, are now based upon order. “Thus, for instance, the smallest of the infinite integers is the limit of the finite integers, though all finite integers are at an infinite distance from it [p. 92].”
Geometry, too, appears in a different light as its non-Euclidean variants have multiplied. Geometry is not about the space in which we live: it is about the valid conclusions that can be drawn from some set of starting principles, where those principles need not accord with what we see in the real world. The leading books in geometry now do not even rely upon figures to demonstrate their arguments. Indeed, figures mislead, as they suggest conclusions that seem obvious from the figures, but do not follow with necessity from the first principles.
Modern geometry does not start with an assumption of some large space; rather, a point is assumed, and then a second distinct point, with lines and other points building upon these beginnings, so nothing is assumed to exist unless it is necessary for the next step of reasoning. The mathematicians responsible for much of this improvement are Peano and Fano. Euclid’s work itself now seems error-filled, in the sense that, strictly speaking, many of his theorems do not follow from his axioms alone. The difficulty of Euclid’s book, along with its errors, renders it unfit for any consideration beyond the historical; it should not be thrust upon English schoolboys. “A book should have either intelligibility or correctness; to combine the two is impossible, but to lack both is to be unworthy of such a place as Euclid has occupied in education [p. 95].” [Russell’s own youthful introduction to Euclid was both intelligible and revelatory. Recall that this chapter of Mysticism and Logic originally appeared in 1901; in 1902, Russell published a short essay with a more detailed critique of Euclid. Even earlier, Russell had dealt with the empirical validity of Euclid’s postulates, an issue he regards in Mysticism and Logic as “a comparatively trivial matter [p. 94].”]
Modern formalism and symbolic logic have brought rigor to mathematics, a rigor that was absent since the Ancient Greeks. Mathematical advances in the interim were so alluring that the foundations of the subject were unexamined. Weierstrass and his ilk were to mathematicians what Hume was to Kant, the prod that awoke a slumbering intellect. Formalism can seem pedantic, but its record in uncovering errors provides its justification.
Kant’s metaphysics cannot survive the fact that mathematics (including geometry) are but elements of symbolic logic: his theory of knowledge was meant to complete Euclid. Now that we know that Euclid is wrong, not simply incomplete, the Kantian theory is not viable. What is needed is for mathematical logic to come to full flower, and then for philosophy to repurpose the same rigorous tools. If this process is successful, the future might see a golden age in philosophy that parallels the recent era of advance in mathematics, and matches the glory that was philosophy in Ancient Greece.
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