To start, let me rehearse Kant’s basic move in explaining our knowledge of the world. Most of us will accept that there are some things we can learn only through experience (like how many donkeys are now in my yard) and some things that are simply true by definition (like anyone with an uncle has a parent with a sibling). Set those aside for now as unproblematic. Kant was interested in a class of statements that are sort of a mix between the two. Take, for example, the fact that any square drawn within a circle will have less area than that of the circle. It has that “true by definition” sort of feel to it, but it isn’t simply true by definition. To see its truth, you need to check it out in your own mind’s eye (“pure intuition,” Kant called it, meaning sense experience unpolluted by anything your physical eye is actually taking in). Call this class of truths “apriori synthetic” facts.
Kant was the first to draw our attention to apriori synthetic truths, and he thought they included many extremely important truths: like the whole of geometry, and the whole of arithmetic, and some super-basic facts about what has to happen in the physical world: that every event has a cause, that when there’s an action, there’s an equal/opposite reaction, and that no event is causally isolated. Indeed, the apriori synthetic truths serve as the foundation for our knowledge of the world.
Now why should it be true that these facts are both known to us through pure intuition, and basic to the fabric of the cosmos? For that is a very nifty coincidence. It could be luck, or God. But Kant’s answer was that the human mind’s own inherent structure makes these truths available to itself through pure intuition, and also puts our experience into a kind of structure so that anything we could possibly experience will fulfill those apriori synthetic truths. They are our operating system, so to speak, regulating both our thinking and all the data that comes our way.
Very cool idea, right? So why did anyone give up on it? It’s hard to give a brief answer to this. I’ll try to write more about this later, but in the end, my view is that Philosophy suffered from a tragic sequence of ideological entanglements. It first had a torrid affair with voluptuous German Idealism; then bounced back from that relationship into a much safer one with Prudish Positivism; then was hit by two buses marked “WWI” and “WWII”, and lost its memory; and finally old Philosophy settled down into a tame relationship with its cousin, Scientific Naturalism. Through all that, the early childhood romance with Kantianism was just sort of forgotten.
There are some solid, worthier reasons not to be satisfied with Kantianism, at least in the form that Kant expressed it. They have to do with the discovery of non-Euclidean geometries and (perhaps) with quantum physics (depending on how you interpret both it and what Kant said about the purest laws of science). And there were some philosophers, the neo-kantians, who tried to repair the old man’s theory; but they too were caught up in various ways with the tragic sequence I described.
But it seems to me that a neo-neo-kantianism is both possible and attractive. This lesson was taught to me a little while ago when I sat in on a physics course. The professor started at one end of the whiteboard with some results we already knew. Then he did a whole bunch of math from that end of the board to the other, and ended up with a predicted consequence. And it turned out that the consequence was verified by experiment, with great precision. Now that is frankly amazing. To ask Kant’s question once again: why should it be that nature knows it has to obey the sorts of mathematical principles we find ourselves bound to think are true?
One might try to say that we learned math from nature in the first place. (That’s naturalism.) But this seems to me quite implausible. For, as I have dabbled in math and number theory, nothing about it seems rooted in empirical discovery at all: it is purely conceptual. Moreover, if you come across a real mathematician who tries to think through to the philosophical foundation of what s/he is doing (and such persons are quite rare), you will have found a platonist. It’s just not natural to think of math as natural. And we know that mathematical truths are not simply the derived consequences of some basic rules mathematicians established at some convention: we have Gödel to thank for teaching us that can’t be right. For any set of rules, there are mathematical truths that cannot be proven by that set – and we can prove that.
So we are left with a spooky coherence between how we have to think and how the world has to be. Why not go Kantian? But this time, we will attribute a different structure to the human mind. It’s hard to say just what that structure is, except to say it is whatever it is that makes math true. Call it MATH, a structure existing among concepts, ordering them and determining how they can be distinguished or combined. I think it may well be that MATH both determines our thinking and the thinking of any intellect; and if it does, then it may well also constrain the data structures in which we try to squeeze the booming and buzzing confusion of our sense experience.
Now MATH is not anything whose existence can be confirmed or denied through efforts in cognitive psychology. For, if MATH exists, any attempt at cognitive psychology must necessarily presuppose it. Thus it would be genuinely apriori, but also synthetic, inasmuch as MATH is not simply true by definition.
So there you have it, my friends: my own neo-kantianism. I cannot lay any claims to its originality, since it may well be close to something Carnap was flirting with in his Logical Construction of the World, but that project got tangled up with a host of other ideological moves I’d rather not make.
Very cool! I believe you are onto something here 🙂
I’m secularly comfortable with this insofar as it is appended to a compensatory nihilistic metaethics.
This blog post was serendipitous for me. I just finished reading Will Durant’s “The Story of Philosophy” and I was disatisfied with how idealism was promptly disregarded without an adequate explanation. That irritated me because of how much sense Kant makes to a philosophy layman like myself (as long as someone other than Kant is explaining it). Your explanation makes sense.
Hmmm. Let me read this about 30 times. Have you said something? Or are you just combining indefinable words in interesting ways? You use use MATH like I use GOD or TRUTH or GOOD. The all caps are useful for indicating that the word is just beyond or way beyond human conceptualization. Is your Neo-Kantian proposal a modern extension of a Spinoza-like MATH=GOD=TRUTH=GOOD?
(Am I arguing for Wittenstein’s position that this is all just word games?) Hmmm.
Good question, Vince. Here is why I think I am saying something. One might think that the world, absent intelligent beings, is the world described by sciences like physics and chemistry. I’m saying it’s quite a bit less than that: it is that world, minus everything relying on the truth of math. I know: that’s a weird world to try to conceive. In fact, I’d say it is impossible to conceive, because of what it is to conceive anything. Once intelligent beings are introduced into the scenario, they start conceiving things, and mathematical structure is imparted to their experience.
Granted, I can’t be very specific about MATH’s intrinsic character; I can just say what function it performs.
Of course one might also object that my claim is not subject to experiment or tests. Right. It is meant as an inference to the best available explanation for the fact that math is so obvious to us and basic to the world we experience.
So neo-kantian is an inverse pythagoreanism (without transmigration of souls) — e.g., not “numbers constitute the true nature of things” but “true nature constitutes the concepts called numbers”.
I might have you wrong here, but I think I disagree. A Pythagorean would say that numbers are the ultimate substance of reality. I’d say something else is – don’t know what – but numbers are at the core of our thinking, and that’s why a Pythagorean might end up believing that they are the ultimate substance.
Are we missing each other’s meaning. Indeed, a Pythagorean would say that numbers are the ultimate substance of reality. I suggested that your proposal is an inverse Pythagorean, that is, reality imposes on to thinking beings what numbers are. A different reality from our reality would have a different MATH. Your neo-kantian proposal being the inverse of the Pythagorean position in a way that makes the Pythagorean position an understandable conclusion.
It all seems to be an unanswerable Chicken and egg situation. Is there only one MATH — and REALITY obeys it (a rational ‘mind’ behind reality)? Or does REALITY impose MATH (reality is the ‘mind’). I use ‘mind’ as an undefined term here, but it implies ‘order’ rather than ‘chaos’.
Did I still miss your point? I am pretty dense with the nuances of philosophy.
What about MATH as REALITY?
When asked about the unreasonable effectiveness of mathematics, Einstein replied: “[A]s far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”
Or, as Richard Feynman put it, “Physics is to math what sex is to masturbation.”
What they might both have been getting at is what Roland Omnès has argued—namely, that mathematics just is physics (or as Roberto Torreti puts in in a review blurb here, that “that mathematical structures are a distillate of the laws of nature”). To paraphrase Nietzsche’s thought on the relation between soul and body, mathematics is just a word about the world.
Er, it’s just a word for something about the world, that is.
Both Einstein and Feynman were referring to math not MATH. I believe MATH would have to be synonomous with reality, or rather, REALITY anyways.
That’s interesting, Michael. Surely there’s a lot of “slop” in physics, from a mathematician’s point of view, since in physics we have to deal with experiments, measurements, margins of error, approximations, etc. It’s not the idealized world of the mathematician (or of the fantasy-ist mentioned by Feynmann!). But the distinction is not as neat or decisive as Einstein’s or Nietzsche’s remarks. In some areas, the math is dead-on, as precise as anyone cares to make it, about how the world works. Certainly in other areas (think of the social sciences, especially economics), the math is at best a shadow of reality. But I’ll follow up on the links you have given!
Well, it’s not necessarily the “slop” factor. (Most scientists would agree that there are tidy natural, mathematical laws underlying all the chaos.) One point is that so far from being “unreasonably effective,” the extant mathematics of the time don’t work at all. Newton invented calculus to deal with problems posed by physics that were insoluble on existing methods. Similarly, when forces couldn’t be understood based on scalars, other mathematical concepts (vectors, tensors) needed to be developed.
I think the anthropological story of mathematics is one where, in response to pressing practical needs, mathematical tools are developed. By their nature are abstractions; and by their methods, the permit derivations. When the practical limitations of their application become clear, new tools are created, new abstractions prescinded, new derivations developed, and (ultimately) new practical limitations discovered. And this picture looks very much like the development of natural scientific theories.
Let’s try again: “By their nature [they] are abstractions; and by their methods, the[y] permit derivations. When the practical limitations of [applying those derivatives] become clear…” etc., etc.
But, Michael, I think you are confusing a small grove of trees for the forest. Certainly there are many interesting cases where math is developed in response to practical scientific needs. But a whopping lot of science, particularly physics, is grinding away with math that was developed independently. Here I’m thinking of basic stuff, like geometry and algebra. I think it would be plausible to say that the reason math matches up with the world in these cases is that the conceptual relations described by the math are apriori, and nature has to conform. But the relations are not analytic, so, etc.
Well, you’re assuming that the development of geometry and alegebra was utterly independent of empirical considerations. As I was suggesting, I’m not sure that that bears out as a matter of anthropology. And I’m not sure why it isn’t just as plausible to say that the reason math matches up with the world is that it was derived, in the first instance, from the world. (“In the first instance” could be extended back to address the origins of innate quasi-mathematical intuitions, which may seem “a priori” as a matter of ontogeny, but are “a posteriori” as a matter of phylogeny—that is, they became innate through the “inductions” of natural selection.)
I think your comment brings us to the hub of the disconnect (egads! now that is truly unforgiveable) between naturalism and some version of rationalism. The rationalism would say no degree of natural induction will yield any necessity and universality to what’s learned. The naturalist might well agree, since he’s not so concerned about necessity and univerality. I think the best thing written on this disconnect is still Leibniz’s New Essays.
Some interesting quotes related to this topic:
Bertrand Russell once said, “Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.”
Hungarian mathematician Paul Erdős, although an atheist, spoke of an imaginary book, in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would exclaim “This one’s from The Book!” This viewpoint expresses the idea that mathematics, as the intrinsically true foundation on which the laws of our universe are built, is a natural candidate for what has been personified as God by different religious mystics. He continued, “Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.”
In Plato’s philosophy there were two worlds, the physical one in which we live and another abstract world which contained unchanging truth, including mathematics. He believed that the physical world was a mere reflection of the more perfect abstract world.
Galileo Galilei is reported to have said, “Mathematics is the language with which God wrote the universe.”
Twentieth-century French philosopher Alain Badiou claims that ontology is mathematics. Badiou also believes in deep connections between mathematics, poetry and philosophy.
I think these Einstein quotes more adequately relate his position on the role of the abstract and the apriori…”Science is the century-old endeavour to bring together by means of systematic thought the perceptible phenomena of this world into as thorough-going an association as possible. To put it boldly, it is the attempt at a posterior reconstruction of existence by the process of conceptualisation…I maintain that cosmic religiousness is the strongest and most noble driving force of scientific research…The whole of science is nothing more than a refinement of everyday thinking…If we knew what it was we were doing, it would not be called research, would it…Where the world ceases to be the scene of our personal hopes and wishes, where we face it as free beings admiring, asking and observing, there we enter the realm of Art and Science…When the number of factors coming into play in a phenomenological complex is too large scientific method in most cases fails. One need only think of the weather, in which case the prediction even for a few days ahead is impossible. Neverthess, noone doubts that we are confronted with a causal connection whose causal components are in the main known to us. Occurrences in this domain are beyond the reach of exact prediction because of the variety of factors in operation, not because of any lack of order in nature.” (This last quote is interesting in light of the somewhat authoratative status given to probability theories over the last 50 years.) “I think that a particle must have a separate reality independent of the measurements. That is an electron has spin, location and so forth even when it is not being measured. I like to think that the moon is there even if I am not looking at it…All religions, arts and sciences are branches of the same tree. All these aspirations are directed toward ennobling man’s life, lifting it from the sphere of mere physical existence and leading the individual towards freedom.”
I think we have to be careful when plucking quotes out of context–clearly Einstein was saying that current conceptions of the world by means of mathematics so not adequately explain reality (our human observations) much less REALITY (or MATH if you will). Nevertheless, such REALITY does exist. (like the weather example, there is an algorythm or point of reference from which it is predictable and orderly–some possible physics–as Strawson would insist.)
(I jacked a bunch of this off wiki etc. but it is interesting to think about the possibility of such connections…)
I am a reductionist in my thought. Probably because I cannot hold too many ideas in my immediate mind at any one time. One of my favorite books is by Etienne Gilson, “The Unity of Philosophical Thought”. It is a favorite not so much in detail but in general concept. He uses ancient, middle age, and modern western philosophers to demonstrate that ‘there is nothing new under the sun’. He did not share the excitement of so many french academics in the avant garde ideas of the latest existential materialists. They were merely repeating the ideas of the Greek schools of thought in the materialism of Epicurus, the extreme cynicism of Diogenes, the skepticism of Pyrrho, etc. There may be new words and new stories, but the basic ideas are essentially the same.
As to the current topic …
I see the basic issue as one of the most ancient of topics even in pre-reason mythos — chaos and order. Chaos should probably be preferred, if anything ‘exists’ at all. If there is something, why is it not in chaos? Why is there order? Mythos says it is a battle between the gods. The gods of order win the battle and impose their order on creation, but there remains a dead or sleeping god of chaos (usually a goddess).
The world has order as we sense the world. Why? Space and time have order and connectedness even if the underlying microphysics has an element of chaos. Each ‘here’ in space is connected to a neighboring ‘here’. Likewise there seems to be a ‘now’ continuously connected to a succession of nows, but these ‘now’s stream past us with a connectedness provided by our memory of ‘past’ nows and our hope of ‘future’ nows. While all ‘here’s are connected in some type of manifold something, they seem to be relating with each other only through the time dimension. Each ‘here’ has a stream of ‘now’s that can be related to other ‘here’s through the a relation of distance and the stream of ‘now’s (physicists identify this ‘local symmetry’). But it all is orderly.
This orderliness — why is it? Why do we perceive it? Why in the world (no pun intended) does abstract mathematics relate to this orderliness?
Now my reductionist view: ‘Orderliness out of chaos’ is the fundamental thing we are trying to explain with philosophy and religion. The essence of nature will always have an undefined quantity at the fundamental ground. Physics will always be left with a fundamental undefined ‘stuff of nature’, but it will be able to describe the orderliness of the stuff! Math arises because of the orderliness. If the orderliness were different, then the math would be different. But why orderliness? Anaxgoras (~500BCE) says there is nous (mind). Cosmic nous is or creates orderliness. ‘is’ or ‘creates’ are the only two positions that I can see. Spinoza (is) or Abram (creates). Socrates (creates) or Epicurus (is). Anaxgoras says that we perceive the orderliness because we also have nous — human mind. We perceive the order with our mind and math arises in our reasoning (logos). In greek and Christian mythology the logos proceeds from the nous or, in other words, math proceeds from nature. There is orderliness (nous) and reason (logos) arises because of the orderliness.
Here is an example of humans developing different math from different natures:
Thales of Miletus took the mystical right triangle mathematics from Egyptian priests and approached the world with an engineer’s mindset. The right triangle ‘laws’ fit nature very naturally (no pun intended). Does the math rule nature or does nature teach us the math? Euclid used 5 axioms and a few undefined words to create the math of Euclidean Geometry. Euclid’s Geometry begins:
“Let the following be postulated”:
1. “To draw a straight line from any point to any point.”
2. “To produce [extend] a finite straight line continuously in a straight line.”
3. “To describe a circle with any centre and distance [radius].”
4. “That all right angles are equal to one another.”
5. “That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”
The 5th axiom can also be stated “Given a line and a point not on the line, then there is one and only one line through the point that will not intersect the line.” (This is not a rigorous restatement because it permits hyperbolic geometries I think.)
It was the stuff of the universe — it seemed. Two parallel lines never intersect. A right triangle follows pythagorean’s theorem. Interior angles of a right triangle add up to 180 degrees. etc. The mathematics of right triangles appeared everywhere in nature. Did nature obey the math or did nature teach us the math? But that 5th axiom always seemed like a provable theorem rather than a necessary axiom. Mathematicians tried to prove that parallel lines never intersect from the first 4 axioms for many years.
Around 1850, Karl Gauss thought about a different 5th axiom: “Given a line and a point not on the line, all lines through the point will intersect with the line.” This yields a different, valid geometry (elliptical geometry) and a different math for right angles. It also reveals a different physics if nature wants to obey ‘non-euclidean geometry’ — which is does in Einstein’s general relativity physics. Math reasoning (logos) was discovered to map nature’s discovered orderliness. Nature is the orderliness which is nous. Math reasoning proceeds from natures orderliness. Do we humans perceive nature because we are from nature (our nous is from nature)? — materialism. Or do we humans perceive nature because we share something with the Nous behind nature (our nous is beyond nature stuff)? — some sort of dualism. Kleiner might disagree with this last statement and say that our human nous can arise a material hylomorphism, but that seems like a quibbling detail to me.
So my distillation summary:
Why orderliness? Is math imposing the orderliness? — Not!
Nous (mind) is orderliness is nature. This is merely an identification of words and not relating our preconceived notions of what a mind is. Orderliness is just orderliness is Nous. An equivalence of words.
Logos (reason or math) proceeds from Nous. This is merely stating that abstract math relationships can be generated to fit orderliness. There are relations within reality because there is orderliness.
There only remains the atheist and theist views of orderliness. Nature is Nous or Nous is behind nature. In both cases we can anthropomorphize Nous and Logos into human mind and human reason, but these are human attributes of perceiving and mental organization of nature via Hume or Kant or whomever. The Nous and The Logos have only analogous relation to human nous and logos (a la Aquinas’s use of analogy in theology).
Kantian additions might clarify that human nous and logos never really discover the ‘Nous in itself’, because our nous and logos is educated through a noise layer of sense perception.
Does neo-kantianism add nuances or overturn my reduction? Is my simpleton logos stuck in my molasses nous.
I embrace just a bit of Wittgenstein symbolic reductionism. We have an intuitive concept of orderliness and chaos.
Nature = Orderliness + Chaos = Nous (Spinoza).
Nature = Chaos. Nous imposes Orderliness on Nature (i.e., Nous over Nature). (Abraham)
Logos = Reason = Math.
Logos proceeds from Orderliness because Orderliness = Nature. (Spinoza)
Logos proceeds from Nous and maps to Nature. (Abraham)
These are all words that gain definition through our human experience of external orderliness and our internal reasoning. Human interaction with Orderliness and Logos is kantian. We have applied our internal perception of our human nous and logos on to Nous and Logos. It is best to accept this mapping of nous onto Nous as analogy only (Aquinas).
The main question is whether Nous = Nature or Nous over Nature. There certainly can be quibbling over which words to use. All other arguments are over the importance of the degree of metaphysical anthropomorphizing of Nous and Logos with the concept of ‘person’.
Vince – thanks for your comment, which draws so many interesting ideas together. As I’ve mulled over my neo-kantianism, I’ve come to some further thoughts that fit in with yours. In overview, it seems to me that my MATH is just a special instance of a more global ordering structure (which I’m tempted to call “IDEA”) which does not merely make our experiences math-friendly, but makes our experiences intelligible, i.e., expressible through concepts. I doubt that there are any apriori synthetic truths legislated by IDEA, but IDEA would be posited as necessary for the possibility of any sort of conceptual inquiry. This has led me back to Ernst Cassirer, whose works I spent a lot of time reading 15 years ago – and what I’m saying may be just what he was saying. I’m still reading and reflecting on it.