Theoretical ambitions

I write here about the sort of theories that I would like to read, and which I would like to write if I were sure of having time and an income. This is basically the stuff that I fantasize about in the shower.

Geometrical method

Social theories should be presented in geometrical style, à la Spinoza, though perhaps with more explanation of definitions and axioms than Spinoza employed, and possibly with answers to objections, à la Thomas Aquinas. This does not ensure that they are more likely to be right, but it makes claims more clear and distinct – whether they are meant as analytic or as synthetic, for instance – and it makes the connections between claims more explicit. Someone who wishes to criticize a claim made by a theory should be able to easily find all other claims on which it depends, and make explicit the stance of his criticism toward all of the dependencies and the validity and soundness of the inferences. This is best done if claims are emphasized and named, and referred to by their names, as is done in the geometrical style, where someone may say, for instance, “this follows from proposition 5 and axiom 2”.

Ethics into law

Natural-law ethics, which is already formulated deductively, should be put into the geometrical form just requested, and elaborated deductively to such a level of specificity that you can actually try particular cases by it. Positive law only exists because ethicists have slacked off.

Of course, I believe that an important practical aspect of law, viz., the measure of punishments, cannot be rationally determined in the abstract. A complete natural-law system of ethics should give some alternative standards of punishment so that the readers can apply the one that seems best to them, or formulate their own standards in relation to the ones given. For instance, historical standards of punishment from various countries could be given, alongside the opinions of notable philosophers, such as the Rothbardian “two teeth for a tooth” standard.

Supposing that there are no legitimate states, and that only legitimate states can apply coercive punishments, the judge who applies natural-law ethics would be determining the proper response of uncoercive punishers, viz., society at large, i.e., the decisions of individual buyers and sellers in the market to boycott or exclude the punished. As such, the judgment of the offenders would be broadcast publicly and each member of society would apply his own standards as he sees fit.

Once natural law can try all cases, its explicit rationality will evidently have more legitimacy than the state, and we may hope for a future where governments are not trusted to make laws.

Economics

Economics should also be developed according to the geometrical method. Since the Austrian School’s tradition already develops economics deductively, the translation should be straightforward, though laborious. Theories from alternative schools could possibly be translated as well.

The system of economics can then be synthesized with the system of ethics into a sound system of political theory, which can guide movements and revolutions. The greater clarity of this system would also allow it to more easily guide the revisionist reinterpretation of historical events; the geometrical system of economics would also be more easily used as a guide to methods in business and accounting.

Voting

 There is no reason to think that voting involves either

• consent to the general system of government, or
• consent to the government of the person you voted for, or
• support for the government you voted for, in any absolute sense.

The only thing which voting inherently involves is support for the government you voted for in preference to the alternatives, since that is the only thing which the act of voting can inherently do. Accordingly, it can only make you responsible for the actions of the government insofar as they are different from what the alternative would have been – and insofar as you could foresee them before voting.

Usually, support for not voting in leftist and libertarian groups comes from misinterpreting the act of voting in one of the three ways I listed above, or something like them. If you cannot, in good conscience, vote without thinking you are doing these things, then by all means don’t vote, but those meanings are entirely in your head.

It is true that voting does almost nothing, and that any time spent on advocacy about voting could probably be better spent on other advocacy, or other kinds of political action. If you live in a country where voting itself takes considerable effort, then the effort is probably better spent elsewhere. To that extent, there is something venerable about the voting-abstinence tradition in these political circles. But no further.

Kant’s categories

Note: This blog post has been retracted, since I no longer think of it as a good representation of how I think about its topic. I may, or may not, have written a better post about the same topic since; check the full list of posts.

I have just finished reading the Analytic of Concepts, which is the part of the Critique of Pure Reason where Kant talks about his table of categories (pictured to the right). I have the following disorganized thoughts about it.

from the Hackett edition; some footnote numbers edited out

The table, undeniably, looks impressive. I appreciate Kant’s attempt to derive it in an orderly fashion, “from a common principle”. But I had several issues with the table as soon as I saw it, and I pored through the rest of the chapter expecting an elaboration of the division and a defense of it; instead, the bulk of the Analytic of Concepts is dedicated to explaining why we should have any categories at all. My hope was lost when I came to this sentence, in which Kant confirmed that no help was coming:

But for the peculiarity of our understanding, that it is able to bring about the unity of apperception a priori only by means of the categories and only through precisely this kind and number of them, a further ground may be offered just as little as one can be offered for why we have precisely these and no other functions for judgment or for why space and time are the sole forms of our possible intuition. (B145–146)

That the categories should even be derived from the functions for judgment in the first place is a very strange doctrine. Categories are fundamental concepts, not judgments; a division of judgments does not seem to bear upon them. Kant seemed to be fascinated by his idea that we can “trace all actions of the understanding back to judgments, so that the understanding in general can be represented as a faculty for judging” (B94). But he does, nevertheless, accept the Aristotelian division of “the higher faculties of cognition” into the “understanding, the power of judgment, and reason” (B169), so he shouldn’t have mixed those up.

He regards the categories as “concepts of an object in general, by means of which its intuition is regarded as determined with regard to one of the logical functions for judgments.” (B128) But they do not seem to actually do this. If the tables are supposed to correspond to each other, then we should ask – can we not think about substance in hypothetical and disjunctive judgments? How are all judgments about impossibility problematic? Kant seems to think that substance is defined as “something that could exist as a subject but never as a mere predicate” (B149) – this seems to regard the concept’s place in a judgment, not the logical function of the judgments where it appears. What gives?

When they do, in fact, determine the logical function, they do not seem to actually be concepts at all; this is the case with the categories of modality. If “existence” refers to something’s being able to be thought, or understood, then it does not seem to be distinct from unity, which Kant says belongs to every cognition of an object (B114). But if it refers to something’s appearing before the senses, then it cannot belong to a concept at all, and belongs rather to the intuition. Besides appearance and thought, there cannot be another sense to the term “existence”; similar considerations apply to the other terms. Only judgments have modality, not concepts. (“Community”, also, is a very strange category, and seems rather to refer to a judgment involving two concepts.)

The division of judgments according to relation is poorly drawn. While these judgments differ in their form of being stated, they do not differ cognitively, since all hypothetical judgments imply categorical judgments. To say that “if A is B, then A is C” implies that “all B are C”, or at least that “all B that are X are C”, where X is something that was understood within the definition of A. The thoughts are equivalent, and only the words aren’t; the division has its place in logic, but not in the theory of knowledge.

Kant says that the transcendental properties of being, “one, true, and good”, are the same as his three categories of quantity, (B113) which I do not dispute. He says, however, that they are “logical requisites and criteria of all cognition of things in general”, rather than belonging to things in themselves. Now, there is nothing more to things than: the things insofar as they are sensed, and, the things insofar as they are understood through concepts. It makes no sense to ascribe anything else to things in themselves, so it is very strange to limit the term like this, as though things in themselves had something more than their sensible and intelligible properties. I would rather say that whatever belongs to things as intelligible (through concepts) is what belongs to things in themselves; indeed, Kant himself seems to do this, when elsewhere he speaks of “the categories, on which nature (considered merely as nature in general) depends, as the original ground of its necessary lawfulness” (B164).

Similarly, Kant restricts the term “cognition” to concepts that have corresponding intuitions, (B144ff.) excluding the categories as “empty concepts”, which are merely “thought” but have no object. This language is strange, since it makes the knowledge of the universal determinants of all possible experience into a lesser form of knowledge. Elsewhere, he refers to the categories as “the manifold of intuitions in general” (B154), and this is quite right, since all intuitions must fall under them; it is more appropriate to say that they have a general object than no object. They do not correspond to anything in particular, but they do correspond to something; properly, only self-contradictory concepts have no object.

For a division of real categories – fundamental concepts which enter, as highest genera, into the formation of all other concepts – see my ontological taxonomy, which is broadly similar to Aristotle and Porphyry’s projects. Kant says that “the objective validity of the categories, as a priori concepts, rests on the fact that through them alone is experience possible (as far as the form of thinking is concerned)” (B125); if we admit that experience must happen through concepts, a thoroughly exhaustive division of categories seems to secure this, since it shows that no concept can be defined which falls outside of them. I believe that my divisions are clearly exhaustive, so that Kant’s elaborate justifications of the categories seem to be superfluous to me.

Kantian terminology

I have already written some posts about Kant, and am probably going to write more. My posts are generally written with my own terminology, which is a reinterpretation of Scholastic terminology. Kant has his own set of terms, however, and those have fairly standard English translations across editions. So, I leave here a note about the differences in terms.

This blog post was updated in 2022-09-28. The update added new sections after the first one, which was given a heading. It also bolded all key Kantian terms that are commented on.

Intuition, pure intuition and imagination

When Kant says “intuition”, he is usually talking about what scholastics would call the senses, including both internal and external senses; accordingly, he calls sense perceptions “intuitions”. He sometimes considers the theoretical possibility of “intellectual intuition”, which would be God’s understanding of things; but he emphasizes that all human intuitions are sensory.

When Kant talks about “pure intuition”, he is usually talking about “constructing” things a priori in it, and so he seems to be talking about the human power of imagination, i.e., of producing sensible images without an object affecting the senses. He sometimes does use the word “imagination”, and he does define it as “the faculty for representing an object even without its presence in intuition.” (B151) But when using the word “imagination”, he sometimes ascribes to it the function of unifying the impressions of the various senses into a single image, which is a function that the scholastics ascribed to a separate faculty, called the “common sense” (sensus communis) or “central sense”. As a footnote to the first edition of the Critique of Pure Reason shows, he seems to be ignorant of this precedent, and to think rather that no one before him had even thought of this function as being separate from sensation itself:

No psychologist has yet thought that the imagination is a necessary ingredient of perception itself. This is so partly because this faculty has been limited to reproduction, and partly because it has been believed that the senses do not merely afford us impressions but also put them together, and produce images of objects, for which without doubt something more than the receptivity of impressions is required, namely a function of the synthesis of them. (A120)

As I have mentioned elsewhere, I tend to avoid the word “intuition”, and I have no use for it in my own work. When talking about Kant, however, I have found it helpful to use it as he did. When I say “imagination”, however, it is always in my sense of the term, which I find to be more precise, since it excludes the central sense. Sometimes, I may speak of Kant’s teachings in my own terms, so that I might speak of “imagination” where Kant had only spoken of “pure intuition”, and I might talk about “sense” in a context where he had said “intuition”. If something was lost in this translation, it was also lost to my mind; I can only think in my terms.

Concept

What Kant calls the “pure concepts of the understanding” would not all traditionally be called “concepts”, i.e., universals, predicated of many things with regard to their essence. The category of community, for instance, could never be part of a definition of what something is. Really, Kant’s categories are better regarded as words that, when placed within propositions, represent “logical functions of judgment”, a phrase which he himself used often.

Kant seemed quite aware that he was using the term in a broader sense, since he said, for instance, that “just as modality in a judgment is not a separate predicate, so too the modal concepts do not add a determination to things” (Prolegomena, §39). However, while I had not read enough of the relevant works, I was thrown off immensely by the use of the traditional name categories, since Aristotle’s categories really were concepts in the traditional sense, and could all be used as parts of definitions of the essence of things. So, I thought that Kant had tried and failed to make a new division of categories in the old sense, and that, inexplicably, he had tried to derive this division from a table of logical functions in judgments, which I thought was very strange. I raised this complaint, among others, in a now-retracted post about his table. Kant should really have just made a new word for whatever he was doing, maybe “judicatures” or something.

Judgment

In that post, I also said the following:

The division of judgments according to relation is poorly drawn. While these judgments differ in their form of being stated, they do not differ cognitively, since all hypothetical judgments imply categorical judgments. To say that “if A is B, then A is C” implies that “all B are C”, or at least that “all B that are X are C”, where X is something that was understood within the definition of A. The thoughts are equivalent, and only the words aren’t; the division has its place in logic, but not in the theory of knowledge.

I think that this opinion is also rather common in traditional logic. Kant actually agreed that “categorical judgments underlie all the others” (Prolegomena, §39). But he apparently really thought that hypothetical and disjunctive judgments are distinct kinds of judgment, which differ from categorical judgments in that they “do not contain a relation of concepts, but of judgments themselves” (B141). That is, they are compound judgments – judgments about other judgments – whereas categorical judgments are simple judgments. This doctrine was apparently shared by Johann Heinrich Lambert’s 1764 work Neues Organon, which I cannot read because I do not know German. Thinking about it now, it seems perfectly reasonable, and I no longer object to it. I mention it here, however, because it means that the word “judgment” was also used in an unexpected sense by Kant.

Cognition and existence

As I mentioned in the retracted post, Kant restricts the term “cognition” to concepts that have corresponding intuitions, (B144ff.) excluding the categories as “empty concepts”, which are merely “thought” but have no object. He also restricts the term “existence” in this way, i.e., to things of which we have a “determinate” cognition; and he seems to do likewise with “knowledge”. I have grown accustomed to his usage, but I still think that it is pretty arbitrary to restrict these words in this way, and that certainly not everyone does so.

In a different post, I suggested that it would be a good thing to apply a “translation key” to Kant, so that whatever he says is “merely thought” “indeterminately” may also be said, more broadly, to “exist” and “be known” (and by implication, “cognized”), while still meaning to express the same philosophical teachings, fundamentally. He makes a very narrow use of very important terms, and I think that his usage is narrower than that of typical speech; in particular, I think that “God exists” is not typically taken to mean that God is given to us as a determinate object in any kind of intuition.

Controverted Questions

Having talked about most of the rest of philosophy to my satisfaction, I resolve here a variety of controversial questions that are very popular, despite being very uninteresting.

0. Contents

1. God
2. Immortality of the soul
3. Nature of time
4. The ship of Theseus
5. Brain-in-a-vat scenarios

1. God

Existence is a very problematic and ambiguous word. In my opinion, it means either that something is sensible or that it is intelligible. Obviously, no one thinks that God is the former, so if the word “exists” is taken in the sense of appearance, as it very often is, then the atheists are right: God does not exist. But if it is taken in the sense of intelligibility, which I identify with being and reality, then God does exist, since God may be understood by a concept; since certainly, the persons who use the word “God” mean something definite by it.

The common meanings of the word God reduce to the notions of, “something absolutely perfect”, and, “the cause of all other things”. In my ontology, both of these things may be said of the most general form, or concept, viz., being, when said without qualification. Since all being is good, and all things are good precisely insofar as they are, being taken by itself must be absolutely good, and much better than any more particular concept. And being is the formal cause of all other forms, since all of them belong to the same genus. So, the “existence of God”, taken in the sense of intelligibility, really follows at once from the principle of non-contradiction, and nothing else can be understood without it. The identification of God with the Platonic “form of the good” is very well-known; since goodness is identical with intelligibility, this is exactly what I’m saying here.

The “attributes of God” are the properties which we believe to be “pure perfections”, i.e., always absolutely better to have than not to have. Since they ascribe no limitation to the things of which they are predicated, they may without falsehood be ascribed to the concept of being, although being does not thereby become a composite concept, since it still means nothing but intelligibility, i.e., understandability through a concept. Since this is not composed of anything else, divine simplicity is also assured.

Regarding religion, the question may be raised of the precise respect in which the blessed in heaven have a superior knowledge of God. At the moment, I suppose that their knowledge is superior in that they are allowed to have an adequate concept of some pure perfections that we cannot understand from our current experience here on earth. So, they know more about what may be said of God. Probably, the more holy and virtuous persons will be granted more of this than the less holy and virtuous.

2. Immortality of the soul

I know of no good arguments for the belief that individual human minds are incorruptible; the arguments that I have seen seem to rely on improperly imaginative thinking. However, I figure that Catholics may believe, by faith in the Catholic Church, that God somehow preserves human minds after the bodies die.

3. Nature of time

Debates in the philosophy of time seem to turn upon the question of whether the past and the future exist or not. Again, existence means either sensibility or intelligibility; I see no other meaning to the word. Now, the past and future are not intelligible as such, since pure concepts have no tense; the forms are timeless and eternal. And the past and future are obviously not sensible. So, they don’t exist right now in any proper sense of the word; in the sense of appearance, it may be said that the past used to exist, and the future will exist, when it happens. This seems to commit me to “presentism”.

4. The ship of Theseus

The question of the ship of Theseus seems to be whether a ship is the same ship after all of its planks have been replaced by other planks. It seems that it does, since throughout each of the replacements, there was still a single collection of appearances (substance) which was understood to be a ship. If the original planks, after being taken out, are used to build another ship made out of the original material, this does not seem to change anything; that ship is a new ship.

It may also be asked whether a ship is the same ship if the new planks are made out of a different material, e.g., metal rather than wood. Here, I figure that it matters whether your concept of a “ship” includes something about the properties of these materials or not. If it is still a ship, then it is the same ship. But if it has come to be understood through a different concept, e.g., if you think that ships have to be wooden, and the new material makes it a boat rather than a ship, then it has become something else, so it is not the same ship, since it is not a ship anymore.

5. Brain-in-a-vat scenarios

I think we have no reason to be absolutely sure that we aren’t brains in a vat, but we have no reason to suspect that we are, either. Either way, the external world that we experience certainly does exist, in the sense that it is apparent, or sensible. It is also intelligible as a whole, though not every part of it is intelligible. There is nothing more to it.

Ontological taxonomy

Having just come up with half of it, I present here a roughly comprehensive division of reality into genera and species.

Tree diagram of the ontology.

First, all things fall under the supreme genus (genus generalissimum), which I call being, and which is defined by intelligibility, i.e., the possibility of being understood by a universal concept. Logically speaking, some appearances fall under non-being, such as defects and vices, etc., but that is only to say that they cannot be understood through a given concept, so that, intelligibly, they are nothing.

What is intelligible can either be imagined by itself, or can only be imagined as part of a whole that includes other things, like how color cannot be imagined without surface. The former are called substances, and the latter accidents.

Substances

Substances, considered as collections of appearances, are either understood to be capable of containing the efficient cause of their motions, or not. If so, they are called living, or animate. If not, then non-living, inanimate.

Animate substances, if they can also contain the formal cause of their motions, are called sensitive, and if not, insensitive. The latter were called plants by Porphyry, but the name seems improper nowadays.

If a sensitive substance, or animal, can also contain the final cause of its motions, then it is rational; if not, irrational.

If a rational substance, by nature, always contains the final cause of its motions, then it is an incorruptible, or immortal, god; if not, then it is a human being, or man. So, the definition of a human being comes out to be, a rational, mortal animal. For ethical purposes, I have found it fitting to consider other features of human beings as we experience them; see the anthropology.

Accidents

Accidents, being defined by not being imaginable by themselves, are divided according to the reason they are not so imaginable.

If they cannot be imagined by themselves because they are proper sensibles, i.e., the proper object of one of the senses – such as color of sight, sound of hearing, and so on – then they are qualities.

They might not be imaginable by themselves because they are common sensibles, i.e., objects common to multiple senses, such as size, figure, etc. As Kant showed in the Transcendental Aesthetic, all such common sensibles reduce to the two pure forms of sense, viz., space and time.

Time, since it has only one dimension, constitutes a species by itself, which is the category called when by Aristotle.

Accidents belonging to the form of space may be divided into size, or quantity; the relative position of a subject, which may be called location, or where; and the relative position of the parts of a subject, which is posture.

An accident might not be imaginable by itself because it is a motion. If it is the beginning of a motion which ends in another substance, then it is an action; if it is the ending of a motion which comes from another substance, then it is a passion; if it is a motion which is already past, but still predicated of the substance, then it is a relation.[*]

Finally, an accident might actually be imaginable by itself, but in fact always imagined as part of another substance, due to an arbitrary convenience, or habit of mind. This tends to be the case with a person’s clothes, for instance; such accidents constitute the category called habiliment, or having.

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[*] Some relations given by Aristotle, such as double and half, really fall under quantity, since no quantity is truly absolute, but they all require comparison. Some relations given by other authors, however, such as parenthood and filiation, are really past motions, and fall under this category.↩

Kant on mathematics

I recently learned about Kant’s theory that mathematics and philosophy are both a priori, but philosophy proceeds through concepts, while mathematics proceeds through the construction of concepts in “intuition” – which, in this context, I believe to be Kant’s name for what I call imagination. This made me think the following thoughts about it. I may have more thoughts later on, or revise these thoughts; I have not finished reading any of the relevant works by Kant.

0. Contents

1. Comparison of Kant’s theory with Ptolemy’s views on mathematical science
2. Interpretation of Kant’s theory, with respect to the necessity of theorems
3. Illustration, drawn from Poincaré, of the foregoing interpretation
4. Comparison of Kant’s theory with Poincaré’s views on mathematical induction
5. Appendix – translation of Lalande, “tautology”, sense B and observations

1. Comparison of Kant’s theory with Ptolemy’s views on mathematical science

Kant’s theory reminded me of the preface that the ancient astronomer Ptolemy wrote to his Almagest, as it may be read in G.J. Toomer’s translation. For copyright reasons, I quote only part of it. Ptolemy comments on Aristotle’s division of theoretical philosophy into physics, mathematics, and theology, and explains his own preference for mathematical studies:

The division of theoretical philosophy which investigates material and ever-moving nature, and which concerns itself with ‘white’, ‘hot’, ‘sweet’, ‘soft’ and suchlike qualities one may call ‘physics’; such an order of being is situated (for the most part) amongst corruptible bodies and below the lunar sphere. That division of theoretical philosophy which determines the nature involved in forms and motion from place to place, and which serves to investigate shape, number, size, and place, time and suchlike, one may define as ‘mathematics’. Its subject-matter falls as it were in the middle between the other two, since, firstly, it can be conceived of both with and without the aid of the senses, and, secondly, it is an attribute of all existing things without exception, both mortal and immortal: for those things which are perpetually changing in their inseparable form, it changes with them, while for eternal things which have an aethereal nature, it keeps their unchanging form unchanged.

From all this we concluded: that the first two divisions of theoretical philosophy should rather be called guesswork than knowledge, theology because of its completely invisible and ungraspable nature, physics because of the unstable and unclear nature of matter; hence there is no hope that philosophers will ever be agreed about them; and that only mathematics can provide sure and unshakeable knowledge to its devotees, provided one approaches it rigorously. For its kind of proof proceeds by indisputable methods, namely arithmetic and geometry.

For a better parallel with Kant, and without violence to the thought of any of these three authors, we may refer to what Aristotle and Ptolemy called “theology” as metaphysics. As seen in this passage, then, Ptolemy seems to have been aware, like Kant, that mathematics and metaphysics are both a priori, unlike physics, which can only be conceived of with the aid of the senses. He also notes, similarly to Kant, that mathematics, unlike metaphysics, treats of visible things, and therefore can make use of the senses. Though predating Kant’s distinctions, he seems even to have come to Kant’s conclusion that mathematics is more evidently an actual body of synthetic a priori knowledge than metaphysics is.

2. Interpretation of Kant’s theory, with respect to the necessity of theorems

Since the imagination is a sensitive power, distinct and separate from reason, I would interpret Kant’s theory as meaning that mathematical judgments are not necessarily true of their objects, but only of their objects as they may be constructed through the forms of human sensible intuition.

These forms, in turn, are essential to human beings, so that mathematical judgments really are necessary truths – but they are necessary to the human nature, not to the nature of the mathematical objects that they involve. A being with different forms of sensible intuition is conceivable, and such a being could make different, and equally true, mathematical judgments about the same mathematical objects.

The interpretation is made clearer by thinking of different axiomatic mathematical systems, which give different interpretations of such human concepts as lines and planes. You could, conceivably, write the same definition of triangle in a non-Euclidean system of geometry that you would use in the Euclidean one, but because the systems interpret the terms differently, it would have different consequences, such as, for instance, its angles not adding up to 180 degrees. This is similar to how a being with different forms of sensible intuition would construct different relations between the same mathematical concepts.

Only the discoveries of deduction and analysis are absolutely necessary to their objects. What we find through our senses is, at best, necessary to our senses – even if we find it a priori.

3. Illustration, drawn from Poincaré, of the foregoing interpretation

Henri Poincaré, in Science and Hypothesis, provides a helpful illustration of the possibility of rational beings with different forms of sensible intuition. He does so by imagining that they live in a world under certain peculiar conditions. Since the work is in the public domain, I quote the relevant section in full:

Suppose, for example, a world enclosed in a large sphere and subject to the following laws:

1. The temperature is not uniform; it is greatest at the centre, and gradually decreases as we move towards the circumference of the sphere, where it is absolute zero. The law of this temperature is as follows:—If R be the radius of the sphere, and r the distance of the point considered from the centre, the absolute temperature will be proportional to R²−r².
2. Further, I shall suppose that in this world all bodies have the same co-efficient of dilatation, so that the linear dilatation of any body is proportional to its absolute temperature.
3. Finally, I shall assume that a body transported from one point to another of different temperature is instantaneously in thermal equilibrium with its new environment.

There is nothing in these hypotheses either contradictory or unimaginable. A moving object will become smaller and smaller as it approaches the circumference of the sphere. Let us observe, in the first place, that although from the point of view of our ordinary geometry this world is finite, to its inhabitants it will appear infinite. As they approach the surface of the sphere they become colder, and at the same time smaller and smaller. The steps they take are therefore also smaller and smaller, so that they can never reach the boundary of the sphere. If to us geometry is only the study of the laws according to which invariable solids move, to these imaginary beings it will be the study of the laws of motion of solids deformed by the differences of temperature alluded to.

No doubt, in our world, natural solids also experience variations of form and volume due to differences of temperature. But in laying the foundations of geometry we neglect these variations; for besides being but small they are irregular, and consequently appear to us to be accidental. In our hypothetical world this will no longer be the case, the variations will obey very simple and regular laws. On the other hand, the different solid parts of which the bodies of these inhabitants are composed will undergo the same variations of form and volume.  Let me make another hypothesis: suppose that light passes through media of different refractive indices, such that the index of refraction is inversely proportional to R²−r². Under these conditions it is clear that the rays of light will no longer be rectilinear but circular.

To justify what has been said, we have to prove that certain changes in the position of external objects may be corrected by correlative movements of the beings which inhabit this imaginary world; and in such a way as to restore the primitive aggregate of the impressions experienced by these sentient beings. Suppose, for example, that an object is displaced and deformed, not like an invariable solid, but like a solid subjected to unequal dilatations in exact conformity with the law of temperature assumed above. To use an abbreviation, we shall call such a movement a non-Euclidean displacement. 

If a sentient being be in the neighbourhood of such a displacement of the object, his impressions will be modified; but by moving in a suitable manner, he may reconstruct them. For this purpose, all that is required is that the aggregate of the sentient being and the object, considered as forming a single body, shall experience one of those special displacements which I have just called non-Euclidean. This is possible if we suppose that the limbs of these beings dilate according to the same laws as the other bodies of the world they inhabit. 

Although from the point of view of our ordinary geometry there is a deformation of the bodies in this displacement, and although their different parts are no longer in the same relative position, nevertheless we shall see that the impressions of the sentient being remain the same as before; in fact, though the mutual distances of the different parts have varied, yet the parts which at first were in contact are still in contact. It follows that tactile impressions will be unchanged. On the other hand, from the hypothesis as to refraction and the curvature of the rays of light, visual impressions will also be unchanged.

These imaginary beings will therefore be led to classify the phenomena they observe, and to distinguish among them the “changes of position,” which may be corrected by a voluntary correlative movement, just as we do. If they construct a geometry, it will not be like ours, which is the study of the movements of our invariable solids; it will be the study of the changes of position which they will have thus distinguished, and will be “non-Euclidean displacements,” and this will be non-Euclidean geometry. So that beings like ourselves, educated in such a world, will not have the same geometry as ours.

4. Comparison of Kant’s theory with Poincaré’s views on mathematical induction

Poincaré also provides a worthwhile comparison with Kant regarding the very main topic of this post, i.e., the type of cognition involved in mathematical reasoning.

Like Kant, Poincaré sought to establish the status of mathematical cognition as synthetic a priori. He did not do so, however, by claiming that mathematics proceeds through construction of concepts in intuition. Indeed, throughout Science and Hypothesis, he seems to use the word intuition in the Cartesian sense, i.e., an indubitable, clear and distinct conception. (This is my interpretation, since he uses the word without ever defining it.) In line with this usage, he actually shares the Cartesian assumption that most mathematical judgments are, in fact, “a series of purely analytical deductions”.

Instead of sharing Kant’s theory, Poincaré defends mathematics as synthetic a priori by interpreting mathematical inductions as being synthetic a priori cognitions, distinct both from deduction and from ordinary (“physical”) induction. As I first explained it in 2020-07-19, in a private Facebook group, Poincaré believed that mathematical induction is distinct in this way for the following reasons.

• It is distinct from deduction because it proceeds from the particular to the general; from seeing that a theorem holds true for a series of a few numbers, we can proceed to the more general theorem. A recursive definition “contains, condensed, so to speak, in a single formula, an infinite number of syllogisms” – while “analytical verification will always be possible” for any particular case, the general case cannot be arrived at purely deductively, because “to reach it we should require an infinite number of syllogisms”.

• It is distinct from physical induction because it is more certain. “Induction applied to the physical sciences is always uncertain, because it is based on the belief in a general order of the universe, an order which is external to us. Mathematical induction—i.e., proof by recurrence—is, on the contrary, necessarily imposed on us, because it is only the affirmation of a property of the mind itself.”

The latter point, of course, amounts simply to saying that it is a priori, which no one doubts; the idea that states of the mind are more certain than sensory impressions also adds to his apparent Cartesianism, which I attribute to his being French.

As I said in the same 2020 post, defending this conclusion was important to Poincaré for this reason:

• This was important to Poincaré because he believed that “the syllogism can teach us nothing essentially new, and if everything must spring from the principle of identity, then everything should be capable of being reduced to that principle.” If this were true of mathematics, then mathematics could not be a science, according to him. “There is no science but the science of the general” – in order for mathematics to be a science, it must arrive at new general truths which are not reducible to its axioms. By distinguishing mathematical induction in this way, he was able to affirm that mathematics, while maintaining “that perfect rigour which is challenged by none”, is yet not “reduced to a gigantic tautology”.

Poincaré thought, then, that by its reliance on mathematical induction at several key points, the system of mathematics secures the fruitful nature of the science.

I disagree with this characterization of analysis and deduction as fruitless and “tautological” – on this, see the appendix to this post. And at this time, I would say that I agree with Kant’s theory, as I have interpreted it, so that all properly mathematical cognitions rely on constructions of concepts in the imagination. Poincaré’s interpretation of mathematical induction seems to follow as a corollary, then, but I disagree with his Cartesian interpretation of other mathematical judgments.

Poincaré’s point about mathematical induction seems particularly helpful, however, insofar as it seems to show us that any purportedly “purely analytic” axiomatic mathematical systems should not be able to make use of mathematical induction at any point. 

5. Appendix – translation of Lalande, “tautology”, sense B and observations

I made this translation of sense B of the term “tautology”, from André Lalande’s Vocabulaire technique et critique de la philosophie, and posted it to a Facebook group in 2021-07-02. I do not understand French. If I recall correctly, I made it by machine-translating the original French and then hand-correcting it according to an edition in Portuguese. (If this seems unreliable to you, then go read the original – there are no professional English translations available.)

TAUTOLOGY, sense B

(Recent sense.) WITTGENSTEIN proposed to call “tautology” any complex proposition which remains true by virtue of its form alone, whatever the truth-value of the propositions which compose it. Tractatus logico-philosophicus (1922), nos. 4, 46 et seq., p. 96.

This usage was widely spread by the Vienna School, in particular by R. CARNAP; see in particular Die alte und die neue Logik, § VII, in Erkenntnis, vol. 1 (1930), p. 2; French translation by the general VOUILLEMIN, Actualitées scientifiques, no. 76, p. 29.

And as, according to the same authors, all the propositions of logic and mathematics present the character of being thus purely formal, and of teaching us nothing about reality, these sciences are called by them “tautological”. See Observations.

Observations

Whether we admit or not the unity of Logic and Mathematics, and the opinion that all the propositions of these sciences are true by virtue only of the definition of their terms, the words tautology and tautological seem to us ill-chosen to express that character. This choice is no doubt explained: first, by the opposition that one wants to mark between the facts of experience, which increase the matter of knowledge, and pure reasoning, which develops its content; second, by the idea, correct in my opinion, that identity is the ideal of logic. But tautology, in fact, implies equivalence, reversibility; now implication, which is the fundamental relation through which mathematics progresses, is a non-reversible, non-symmetrical relation: a ⊃ b does not necessarily entail b ⊃ a ; and when both are true at the same time, this is vi materiae. The result of a deduction is, therefore, far from being identical, even in pure theory, either to the set of its premises, or to any one of them. It cannot even be said, in many cases, that there is “partial identity” between them. See “Logique et Logistique”, Revue Philosophique, January 1945, p. 76. (A. L.)

On the other hand, G. Bachelard writes, “if mathematics is tautological, why is it so varied, so difficult, so interesting?” cf. “Logique, mathématiques, et connaissance de la réalité”, in the Recherches philosophiques, 1935–1936, p. 450.

Same observations from A. Bridoux, A. Burloud, G. Davy, Ed. Le Roy, D. Parodi.