So Kant concluded vs. the Second Antinomy that matter is indefinitely divisible, so he would have taken issue with the idea that the Planck scale is the absolute limit, here. At first, I was thinking he would have favored a preon model (and indeed beyond this), but another option I've considered is leaning on the analogy between scalar fields and fractals, and saying that the metrodynamic character of a scalar/quasi-fractal structure is good enough to "fill space to infinity," thus satisfying Kant's other no-perfect-voids parameter (his argument was something like: a perfect (physical) void would be void of causation from itself; so it could not possibly cause us to perceive itself; so it is not an object of possible experience).

However, the technical description of scalar fields, if I understand what I'm reading, is more like a field of intensive degrees. The analogy/example I've seen is a temperature map. But intensive degrees, in Kant's pre-model of the universe, are originally located in the anticipations of perception. He does talk about degrees of consciousness vanishingly smaller and smaller to potential infinity, and seems to correlate this issue with apperception's zero-dimensional reality vs. external physical space, yet at the same time allowing that the soul (as the an sich of apperception) "might, for all we know" be capable of diminishing to nothingness intensively nevertheless. Now QFT in general seems like it could be framed in at least neo-Kantian terms as carrying through strongly the third analogy of experience (of the community of substance), so then quantum scalar fields moreover would be joint applications of the anticipations and the analogies, perhaps.

Still, then, where has Kantian matter gone? I am tempted to say: he only says that matter is indefinitely divisible, though, after all. I wish I could remember the name, but I think there was a physicist who actually did think as such: for him, the limit imposed by the Planck scale was not objectively absolute through and through, but simply the place we had reached in our descent to now, and ought not be thought of more strictly than that (although strictly so within the confines of our present theory, even so).

Kant's indefinitude claim overall In the SEP article on Kant's critique of metaphysics, it says at one point:

Obviously, the success of the proofs depends on the legitimacy of the exclusive disjunction agreed to by both parties. Both parties, that is, assume that “there is a world,” and that it is, for example, “either finite or infinite.” Herein lies the problem, according to Kant. The world is, for Kant, neither finite nor infinite. The opposition between these two alternatives is merely dialectical. In the cosmological debates, each party to the dispute falls prey to the ambiguity in the idea of the world.

Another SEP article on infinitesimals says:

And for appearances, Kant maintains, divisions into parts are not completable in experience, with the result that such divisions can be considered, in a startling phrase, “neither finite nor infinite”. It follows that, for appearances, both Thesis and Antithesis are false.

Later in the Critique Kant enlarges on the issue of divisibility, asserting that, while each part generated by a sequence of divisions of an intuited whole is given with the whole, the sequence’s incompletability prevents it from forming a whole; a fortiori no such sequence can be claimed to be actually infinite.

In the first Critique proper, the issue is actually more nuanced when it comes to the divisibility of matter. At first, Kant outlines a description of the divisibility regress that does not proceed ad indefinitum, but outright ad infinitum. However, he goes on to maintain that we are not given an empirically real infinite set of parts of matter, and in the presentation of the Second Antinomy he discourses on the propriety of the phrases compositum reale and compositum ideale with respect to the mereological structure of space, favoring the latter as a vague possibility but holding that, with respect to empirical reality, the former is simply incorrect (space is not an absolute whole in itself, but as a relative whole in itself it is prior to its given parts). So the empirically real elements of the regress are indefinite in quantity, or in other words there is no absolute division of matter obtainable at any moment in empirically real time, so there is no basis for holding that we have ever divided matter into a finite number of parts beyond which further division is metaphysically impossible. There is no basis for holding that there are an actually infinite number of parts of matter given in experience, even if the transcendental question from which the divisibility regress emerges is an authentic object of reason and presents to transcendental reflection as infinite in itself. {Hence Kant goes on elsewhere to refer to the focus imaginarius, as a sort of 'real question' built into the a priori mind, but which, thought of as a cosmological wh-term in the mind, can never be assigned an infinitely complete semantic value.}

  • -1: for opinion dressed up as research. Apr 5, 2022 at 16:08
  • The second antinomy is a contradiction -not an statement- about the indivisibility of things. If fields are things, their divisibility is contradictory.
    – RodolfoAP
    Apr 6, 2022 at 23:03

3 Answers 3


@Kristian Berry Your question touches two important issues. They deal with the relation between philosophy and science in explaining fundamental questions of our worldview. Nevertheless, for me your text mixes up two issues which better stay in separation:

  1. What does Kant want to show with the second antinomy?
  2. Which models are discussed in contemporary physics for the conception of space?

Ad 1) As @PhilipKlöcking has shown there are some snares in Kant’s formulation. They prevent a quick understanding of what Kant means and against which positions he argues.

For a better understanding of Kant’s text one needs an accurate comment. It has to show which concepts of his forerunners Leibniz and Wolff Kant references on the philosophical side of the antinomy. And which Newtonian concepts Kant references on the side of science.

Ad 2) The concept of a scalar field is totally secondary for the concept of space: You name as example a temperature field, you could also name the field of atmospheric pressure on earth. The simple characteristic of a scalar field is that the physical quantity in question depends on 1 (one) parameter, but not on 2 or 3 or …

Special Relativity conceptualizes space as one part of the 4-dimensional spacetime. And General Relativity treats the latter as the gravitational field.

A different view on spacetime has been favoured by Wheeler’s geometrodynamics. The latter has been further developed by Rovelli’s formulation of loop quantum gravity. In loop quantum gravity space is quantized on the Planck scale and treated by methods of quantum theory.

Concerning matter: An ongoing splitting of matter to find smaller and smaller units of matter is impossible: The decomposition of elementary particles needs high energies. These energies generate new particles which are not smaller than the original ones.

  • It was something of a toss-up between this and PK's answer as the best, but my only comment at this point will be to say that I probably emphasized the wrong part of the Critique in formulating my question. I think that maybe I should have focused on Kant's arguments concerning empty space. Then again, cosmological constants would already harmonize with a "no-empty-space" thesis, so bringing up scalar fields or interlocking 3D fractals might not be very useful, in that context (AFAIK the dark energy field might be a 5th force field, or it could be the other fields vs. vacuum energy). Apr 6, 2022 at 20:02
  • "These energies generate new particles which are not smaller than the original ones" Preons are smaller than quarks and leptons, or W- and Z-bosons, or Higgs particles. Smaller than preons it can't get...
    – Pathfinder
    Apr 10, 2022 at 16:17
  • @Felicia What about the state of the art: Have preons be detected in the meantime?
    – Jo Wehler
    Apr 10, 2022 at 16:29
  • 1
    @JoWehler Well, you could argue that the muon g2 experiment has seen sub structure, but this is not considered as evidence. The standard is still that quarks and leptons are elementary. That's the prevailing icon. But why not? The small distances haven't been probed yet. Would be nice to smash electrons head on. But that's not done because of...the icon. So experiments to clash the icon are not to be expected.
    – Pathfinder
    Apr 10, 2022 at 16:40

Sorry to say, but your reading is flawed or at least not based in Kantian thought, as it is contrary to his intentions. The difference ad infinitum and ad indefinitum is introduced in the context of discussing causal chains. He clearly writes later:

It is entirely otherwise with the problem how far does the regress extend when it ascends from the given conditioned to its conditions in the series: whether I can say here that there is a regress to infinity or only a regress extending indeterminately far (in indefinitum), and whether from human beings now living I can ascend to infinity in the series of their ancestors, or whether it can be said only that as far as I have gone back, there has never been an empirical ground for holding the series to be bounded anywhere, so that for every forefather I am justified in seeking, and at the same time bound to seek, still further for his ancestors, though not to presuppose them?
To this I say: If the whole was given in empirical intuition, then the regress in the series of its inner conditions goes to infinity. But if only one member of the series is given, from which the regress to an absolute totality is first of all to proceed, then only an indeterminate kind of regress (in indefinitum) takes place. Thus of the division of matter (of a body) that is given within certain boundaries, it must be said that it goes to infinity. For this matter is given in empirical intuition as a whole, and consequently with all its possible parts. Now since the condition of this whole is its part, and the condition of this part is a part made of parts, etc., and in this regress of decomposition an unconditioned (indivisible) member of this series of conditions is never encountered, not only is there nowhere an empirical ground to stop the division, but the further members of the continuing division are themselves empirically given prior to this ongoing division, i.e., the division goes to infinity. (A512|B540 f., bolded of complete sentence mine).

In other words: He uses the difference between ad infinitum and ad indefinitum exactly to contrast the infinite division of the totality of an object of matter (regress of decomposition as he calls it explicitly in A514|B542) with the indefinite regress of a causal chain of which we can possibly only encounter a limited number of members as empirical objects.

He repeatedly clarifies that speaking of empirical objects we can decompose them indefinitely and speaking of empirical causal chains we may theoretically be able go on indefinitely but always only have a causal chain in indefinitum available to intuition and should not say anything about them beyond that as this would be an illegitimate application of a mere rule of reason to objects themselves like monadists were doing.

The main reason why the decomposition has to be infinite is that we remain in the realm of experience here and decompose a given totality of a manifold of intuition that is infinitely richer than any sum of its empirical decompositions, therefore a totality of which we can therein find ever more parts "because no experience is bounding absolutely" (A514|B542 f.) You may notice that I use "bounding" instead of "bounded" like in Guyer's translation here. The reason is simple: His translation is wrong. Kant writes "weil keine Erfahrung absolut begrenzt", which is active and makes perfect sense as that is exactly the reason for why we can continue to search for further elements of a whole ad infinitum in this kind of decomposition.

This is also seemingly problematic when we think of the famous §77 of the Critique of the Power of Judgement where he plainout denies that we are even able to apprehend objects other than via its parts, unable to think their totality. This contradiction can only be lifted if we assume that there, he was not writing about objects of experience (empirische Vorstellungen) but objects themselves, which makes perfect sense if we consider the context and exact wording of both statements. Accordingly, this constitutes another argument in favour of my reading.

This kind of decomposition, which is literally finding new aspects in the infinitely rich manifold of intuition apprehended as the totality of an object of experience, is - or at least can be - very different from the "parts" of matter that modern physics try to describe. Additionally, QFT is definitely a mathematical description and Kant warns us, repeatedly, not to mingle infinitesimal mathematics with metaphysics and urges to keep experience and theoretical possibilities apart from one another carefully. He stricly speaks about conceptual decomposition of empirical givens here. Probably we could describe it in modern terms to be closer to a phenomenology of ontic being (the world as it discloses itself to us), which includes, but amounts to more than physics. Thus, I say that the SEP article on infinitisimals cited is misleading at best and your conclusions are not backed up by the original text. Maybe another case of "lost in translation", which is often the case when it comes to Kant's texts.

  • I appreciate the translation issues you bring up, I was worried about those in the background. But my stronger claim, here, would involve saying that Kant did not fully understand all of the concepts he was using, the case in point being the finite-indefinite-infinite trichotomy, which I think is technically incomplete, though Kant was not, apparently, in a position to recognize the gap (he almost closes in on the idea of transfinite numbers when he forms the theory of the categories, but then forthwith denies that infinity can be numerical in the required way). Apr 6, 2022 at 10:17
  • My understanding is also that Kant denied the existence of perfect physical voids, and used this as a premise in the presentation of the Second Antinomy later. Pure matter that could not be empirically divided but could be a priori divided across empty interior spaces was ruled out, I thought, so that simple substances were likewise ruled out. Apr 6, 2022 at 10:19
  • But so I wondered if scalar fields might be another mathematical development that answered, indirectly, to one of Kant's theories, not quite to discredit him or conflict with him, but simply to show another solution to a formal question he legitimately posed, and gave a valiant enough attempt at a solution of his own to. Apr 6, 2022 at 10:21
  • 1
    @KristianBerry Did some edits, do not know whether they clarify something for you. I would still hold that scalars and field theory are "of the mathematicians", a theoretical construct. Of course, we could trivially say that any experimentally validated solution to the equations available to us suggests that we can only divide indefinitely, but that is beside the point: Kant uses the difference because of the indefinite richness of a manifold of intuition vs. a causal chain with "quantified" parts (which are totalities, but irrelevantly so in this treatment) which we always only know a part of
    – Philip Klöcking
    Apr 6, 2022 at 10:32
  • 1
    Thus, while him coming up with the term indefinite is certainly interesting, we should not use that term in this application imho, or if so, only to illustrate the difference between indefinite and infinite.
    – Philip Klöcking
    Apr 6, 2022 at 10:34

So Kant concluded vs. The Second Antimony that matter was infinitely divisible ...

He did no such thing. The SEP states he sets out an antimony: thesis, that matter is atomic in the original sense of this term; that is, it is composed of some ultimately simple parts and antithesis: that matter is infinitely divisible.

And again according to the SEP, he concludes neither is true, the truth lies elsewhere.

... so he would have taken issue with the idea that the Planck scale os the absolute limit

No, he would not have. If anything, the notion of the quantum is Kantian: waves are infinitely divisible, particles are not. So he has in essence his antimony here which is partially resolved in a synthesis called wave-particle duality. A truth which is, to be precise, neither one nor the other and is ontically new. This is what Kant suggested and this has been bourne out by scientific fact.

I wish I could remember ...

Perhaps you could begin by remembering to check what the actual literature says?

  • LOL you didn't even quote me right. I said "indefinitely divisible," not infinitely so. The whole method of resolving the mathematical antinomies revolves around this distinction. Apr 5, 2022 at 17:46
  • @Kristian Berry: So am I to correct your mispellings too? Indefinitely divisible is the same as infinitely divisible. The jokes on you and it isn't even that funny. Apr 5, 2022 at 18:08
  • Kant: Very different is the case with the problem: "How far the regress, which ascends from the given conditioned to the conditions, must extend"; whether I can say: "It is a regress in infinitum," or only "in indefinitum" ..." He does identify a form of the divisibility regress as properly infinite, but the description of what can be admitted as empirically real still satisfies the indefinity parameters for the first antinomy, indeed the seemingly same parameters he uses in resolving all the antinomies (even causal regression is not absolute, I think). Apr 5, 2022 at 21:18
  • @Kristian Berry: Pasting and copying stuff does not demonstrate understanding. Apr 6, 2022 at 0:29
  • OMG, I am literally using the same wording as Kant, in the same context, for the same purposes. Your fantasy that I am somehow misspeaking about this topic is bizarre to say the least. I can understand having various answers to my OP or questioning my interpretation of the conceptual situation in the first place, but the hostility involved in this case is absurdly inappropriate to this site. Apr 6, 2022 at 0:38

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