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.

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.

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…

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.

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.

Mysticism and Logic, Chapter IV

"The Study of Mathematics," pages 58-73

Deep contemplations lead to beautiful edifices, but that beauty is remote to the beginner and its achievement and appreciation require hard-to-obtain knowledge. “Dry pedants possess themselves of the privilege of instilling this knowledge: they forget that it is to serve but as a key to open the doors of the temple [pages 58-59].” Their students see only the steep upwards path, and not the gorgeous structure at the end.

In terms of concealing the ultimate purpose, mathematics education might be in even worse shape than classics. The significance of mathematics is often couched in how it leads to better machines, improved transport, and military might. Of course, the limited mathematics training that most people receive does not conduce to these ends. Why do they study mathematics? The typical response is that the study of math enhances the ability to reason – though this response is made primarily by people who themselves teach all sorts of fallacious nonsense. Improved reasoning itself is viewed as contributing to prudent personal decision-making: hardly a goal worthy of teaching mathematics to all educated people. But Plato understood that mathematics is requisite for the apotheosis of mankind.
 
Mathematics pairs truth with unvarnished beauty; it offers respite from the painful compromises of our quotidian existence. The beauty of mathematics is the result of rigorous logic, not a product of any conscious aesthetic design.

How can this beauty, this higher quality of mathematics, be communicated by teachers? In geometry, it is best to focus, at first, not on theorems and proofs, but on diverse illustrations (of triangles and their associated lines, meeting in a single point, for instance). “In this way belief is generated; it is seen that reasoning may lead to startling conclusions, which nevertheless the facts will verify; and thus the instinctive distrust of whatever is abstract or rational is gradually overcome [p. 62].” The concrete is the way to the general.

For new learners, algebra is hard to comprehend, with its reliance on letters (variables) instead of numbers. But algebra presents truths that go beyond the particular, general truths, and it is these that allow “the mastery of the intellect over the whole world of things [p. 63].” Teachers tend to fail to impart an understanding of the principles brought to bear in algebra, however, even if students learn recipes to apply rules that produce correct answers.

After algebra, it is dealing with infinity (as in the infinitesimal calculus) that presents the next hurdle. “The solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our own age has to boast [p. 64].” Cantor and Dedekind showed that when dealing with infinity, you could remove some elements from a set, and still have the same number of elements in the set. This notion cleared up all of the misconceptions concerning infinities, and has opened up grand new vistas of thought. In the past, much of the foundation of mathematics was clearly fallacious, a practical compromise that mixed logic with superstition; now, the need to compromise has been expunged. Pure mathematics, mathematics as an end in itself, can be built from first principles that themselves can survive intense scrutiny.

Textbooks typically fail to convey the unity and purposeful progression of mathematics. But the beauty and drama of mathematics lie in its interconnections, in the relation of many propositions to a few fundamental ideas. Learners must not be distracted from these core notions through a plethora of inessential or unconnected examples.

The ultimate unifying discipline within mathematics is symbolic logic, which is a product largely of the nineteenth century and still developing today. “The true method of discovery in symbolic logic, and probably also the best method for introducing the study to a learner acquainted with other parts of mathematics, is the analysis of actual examples of deductive reasoning, with a view to the discovery of the principles employed [p. 67].” The realization that only a few simple concepts underlie mathematics, and that those concepts are regularly (but perhaps unconsciously) employed in thinking, can be transformative. What had been glimpsed is now seen clearly, and the panorama is beautiful.

Symbolic logic has replaced the former foundations of mathematics, which were philosophical, and thus uncertain. The sounder base has rendered the superstructure more intellectually pleasing, and this pleasure should be made available to students.

Logic and mathematics exist outside of humans and their thoughts – but we can still appreciate the beauty of mathematical objects, whether they be our creations or our discoveries.

For students, the goal shouldn’t be just to inform them of conclusions, of the end points of a chain of reasoning, but also to take them along the most splendid path to those ends. “An argument which serves only to prove a conclusion is like a story subordinated to some moral which it is meant to teach: for aesthetic perfection no part of the whole should be merely a means [p. 70].” Elegance and generality in a mathematical argument, a proof, derive from using only the most fundamental, minimal assumptions necessary to reach the conclusion.

The common notion that truths are relative, that one person’s truth need not be another’s, and that there is no impartial standard to decide the matter, meets its demise in the arena of mathematics.

Mathematics needn’t rely on its practical effects for its justification. But in a world full of injustice, sometimes it seems hard to countenance a life spent in thought, aloof from the evils of the world, while enjoying a beauty that is not available to most people. Of course we need some people to “keep alive the sacred fire [p. 72],” but this rationale might seem inadequate in the face of current troubles. Here, the practical applications of mathematics can help, by reminding us that cloistered study in the near term can lead to tremendous improvements in human happiness down the road. Could we harness steam power or electricity without the development of mathematics? Of course, we cannot know what sorts of mathematics will lead to the best innovations in the future, so we should avoid investing solely in those branches of mathematics that have proven useful in the past.

The love of truth holds the power to raise our moral existence, “and in mathematics, more than elsewhere, the love of truth may find encouragement for waning faith [p. 73].” Cloistered study is its own reward, but it also ennobles our minds; the teaching of mathematics should keep in mind this indirect benefit.

Mysticism and Logic, Chapter III

"A Free Man’s Worship," pages 46-57

Russell begins with a lengthy quotation from Faust, where Mephistopheles explains how the Creator, bored with the praises from angels whose situations he had made joyous, decides to allow worlds and living creatures to develop – living creatures who can be led to worship Him even though they receive only torment, not joy. Eventually men succeed in this task, and even name as Sin the pursuit of their natural instincts. So the Creator destroys these worlds, that He might enjoy a similar pageant once again from the beginning. Russell endorses the fable: “Such, in outline, but even more purposeless, more void of meaning, is the world which Science presents for our belief [p. 47].” We must accept what our best knowledge makes all but certain, that what has been built has been accidental, unplanned, that the grave is our full ending, and that a similar grave awaits the solar system.

“What though the field be lost?/All is not lost.” The interim is ours (and perhaps ours alone), “to examine, to criticise, to know, and in imagination to create [p. 48].”

Savages, knowledgeable of their vulnerability to nature, worship power, even being willing to shed much blood in a fruitless attempt to appease brutal gods. The size of the sacrifice is indicative of the slavish commitment to suppressing any doubts as to the worthiness of the worshiped idol. They cannot conceive of a better world, so they worship what is powerful, not what is benevolent. Today’s might-makes-righters, including social Darwinists and militarists and followers of Carlyle and Nietzsche, are similarly in thrall to power.

The standard religious view is now different – it asserts that our world in fact comports with the benevolence of the all-powerful creator, but in a manner that is unfathomable to mortal minds. This claim is not easily reconciled with the fact that the world contains much evil. We must choose whether to accept that God is evil, or whether to accept that God is unreal, our own invention.

To worship strength is to sacrifice our ideals to mollify the powerful. If strength must be idolized, why not choose to idolize the strength shown by those who are willing to recognize that “is” does not imply “ought,” and the fact that there is much wrong with our world? Our actions must respect the unavoidable constraints of an imperfect world, but our vision – the vision that motivates our actions – should remain fixed on what is good.

The alternative of taking umbrage at the evil of the world is not a real alternative at all: it allows evil to preoccupy us. “Indignation is a submission of our thoughts, but not of our desires; the Stoic freedom in which wisdom consists is found in the submission of our desires, but not of our thoughts [p. 51].” Art and philosophy emerge from the visions of beauty that are licensed by unconstrained thoughts, thoughts uncontaminated by the need to serve our desires.

Austere religions are right, more often than not, in their contention that what is desired is bad. Even those things that we desire that really are beneficial should not become fetishized, if they cannot be attained. And though it is not understood by young people, those who are older know that many wonderful things will be unattainable. “What fates impose, that men must needs abide;/It boots not to resist both wind and tide.”

Along with our renunciation of desire, we need to make it possible to deify our ideals, through our embrace of reason and culture. “In all the multiform facts of the world – in the visual shapes of trees and mountains and clouds, in the events of the life of man, even in the very omnipotence of Death – the insight of creative idealism can find the reflection of a beauty which its own thoughts first made [p. 53].” Tragedy is the queen of the arts in that it takes evils such as death and despair and fashions beauty out of them. Even in the recognition of loneliness, loss, and powerlessness, we can forge a bond with the world, and overcome the sway of these potent adversaries.

The past’s beauty lies in its fixity and purity. “The Past does not change or strive; like Duncan, after life’s fitful fever it sleeps well; what was eager and grasping, what was petty and transitory, has faded away, the things that were beautiful and eternal shine out of it like stars in the night [p. 55].” The past’s beauty can provide a sort of religious rapture.

Appreciating the force of nature in its tragic form is a source of power and freedom. “To abandon the struggle for private happiness, to expel all eagerness of temporary desire, to burn with passion for eternal things – this is emancipation, and this is the free man’s worship [p. 55].”

We are united in the tomb, our common end. As we march along our path, our comrades (along with us, eventually) fall off the cliff, lost forever. We do not have long to ease their steps, to light their path. We shouldn’t concern ourselves with their desert, but with their need. They, too, are fated to be players in a tragedy. When they pass into eternity, let us hope that we did not stoke their sorrows, but rather relieved their pains and enhanced their joys.

Old Man River keeps on rolling; we can be proud in our temporary defiance, as we briefly construct and maintain a world based on our ideals.