Math, matter, monism, and dualism

Metaphysics & Epistemology
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(I have commented on this in other threads, but I figured that my ponderings on the subject deserved their own thread.)

Most views of reality can be divided into monism, as represented by physicalism and idealism, and dualism, as represented by the likes of substance dualism, property dualism, and Platonic dualism.

However, none of these views seem to really neatly incorporate the view that math forms a realm either more fundamental than matter or equally fundamental to matter, that mathematical constructs are not merely human creations or approximations of an underlying reality, but rather that the underlying reality is implemented in terms of math.

If one has not read it yet, a good piece on this topic is Eugene Wigner’s The Unreasonable Effectiveness of Mathematics in the Natural Sciences. It discusses things such as how on very scanty evidence scientists have made predictions based almost entirely on math that hold perfectly so far as we can tell, not breaking despite our best efforts to find counterexamples (and when they do break, that is only because there is math that even better describes reality, cf. the supersession of Newtonian physics by relativity), and that mathematical concepts that were created by mathematicians simply due to being cute ideas, without any intention of relating to the physical world, then got pressed after the fact by scientists into service to describe the physical. By this, it leads one to the conclusion that math has some kind of special connection with the universe that could not be true if it were only a human approximation of reality or merely a beautiful intellectual toy.

As for the independent existence of math from physical reality, the reason to believe in this is that mathematical ideas hold no matter what physical terms they are expressed in. For instance, the value of pi is the same for all circles, no matter how those circles are expressed in the physical substrate or if they are entirely abstract, having no physical reality in the first place.

If one truly believed in physicalist monism, one would have to explain the above, how math appears to be more than merely a human approximation of physical reality, how math appears to be both independent of physical reality and simultaneously fundamentally underpinning it.

The only way I can see how to reconcile physicalist monism with this is to posit that math is more fundamental than matter, and that our physical reality itself is a mathematical construct, rather than positing math and matter as two separate realms.

Note, however, that while many mathematicians have classically been Platonic dualists, Platonic dualism itself posits that ideal Forms exist outside the physical realm that the physical realm approximates, while here I am making no assumption that ideas like “Beauty”, “Justice”, and “Goodness” have any independent basis whatsoever. If anything, these things arise from human existence in the material realm rather than having any basis in the ideal.

Of course, one could limit Platonic dualism so that only things that can be represented in mathematical terms have any reality independent of the physical. This is closer to my position here.

I personally dismiss Cartesian mind-body dualism out of hand, mind you, because to me there is no reason to believe that sentient beings should have any privileged status given the laws of the universe, and because it is clear that the physical impinges on the operation of the mind (e.g. the effect of psychoactive substances) in a way that would be hard to justify if the mind were a fundamentally distinct quantity from the physical body.

So fundamentally I run into two different philosophical positions from which to choose — either a version of monism where physical reality is math itself, or a version of Platonic dualism where the only things that can exist independently of physical reality are things that can be expressed as mathematical constructs of some sort or another. I am not sure which is more valid overall.

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I have a strong intuition, you might want to call it a prejudice, that mathematics is a language, created by humans, that can be used to describe our world in very precise ways. I’ve thought about it quite a bit, but I don’t have good answers to all the points you’ve made. I do have a few thoughts.

Math is “unreasonably effective” only in a relatively narrow range of simple situations—those where basic independent elements can be isolated from the surrounding environment. This is the realm of fundamental physics and, generally, conditions far away in scale from everyday human life—very small and very big things; neutrinos and the big bang. At scales closer to our world, things are more likely to follow statistical rules more closely than algebraic ones.

As an engineer, I’ve seen how much of how we do things is a patchwork of math, fudge factors, and safety factors. Land surveying, for example, where precision is very important, achieves this goal with a set of corrections and work arounds that scramble to stuff the pieces of our unruly universe into Euclid’s rigid geometry. To be fair, it does a very good job.

And then there is chaos theory and quantum mechanics, which seem to put the lie to the idea of a mathematically predictable universe.

I acknowledge I don’t find these arguments to be fully satisfying responses to the points you’ve made and I don’t expect you to be convinced. I’ll be interested in seeing what other people have to say.

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Interestingly enough, math is most effective at describing realms most removed from the everyday human experience, e.g. your examples of the very small and the very big, while the areas in which it is not quite as effective are those which are fundamentally tied to the everyday human experience, e.g. the social sciences.

If math were merely a human approximation of physical reality, a mere pragmatic tool, I’d say you would expect the opposite pattern, that math would be most accurate close to the human experience and less accurate as one goes further from it.

This implies that math’s ties to physical reality are strongest at the most basic levels, where fundamental physical laws have an unequivocable role in all behavior.

This, of course, is because the surface of the Earth is not a perfect rectilinear plane or sphere, which is not in contradiction with physical laws being based fundamentally in math.

Note however that these do not contradict math having a privileged role in the universe, as we can mathematically demonstrate how things are uncertain, rather than them being merely arbitrary, with no rhyme or reason.

As I mention above, the key thing is that the further you get away from fundamental physical laws the less directly applicable simple, raw math gets. But that does not mean that said fundamental physical laws are not mathematical in nature.

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I have read Wigner’s essay, and several other sources on this question.

I agree that number is not constructed by the mind (constructivism) or that they’re useful fictions (fictionalism). I agree that number is real - but in what sense?

The heuristic I’ve developed is that intelligible objects are real but not existent. They’re real, in that arithmetical primitives (the rudimentary elements of arithmetic, such as the natural numbers) are the same for anyone capable of counting them. But then there’s the question of the sense in which they exist.

You can’t point to a number. You can point to a numeral, but that is a symbol, and the same numerical value can be represented by diverse symbols (7, seven, VII, etc.) But the meaning of the symbol is invariant and precise, a real value. To put it simply, the numeral or sign is the phenomenal indicator, what appears to the eye (or the mind’s eye). But the meaning of the symbol is purely intellectual, i.e. it is a value that is grasped by the mind — any mind that is capable of counting, not your mind or mine, so not subjective.

However to elucidate further, I will first respond to this:

I think this is rather a caricature of Platonism. And it’s central to grasping the role of mathematical Platonism, you can’t carve this off without mutilating the entire system of thought.

While it’s true that Plato himself speculated about the ‘topos uranus’, a ‘special place’ where the Forms resided, this is really a poetic analogy. The form (eidos) is better depicted as a principle or the essential nature of a particular.

Plato Forms

Plato’s metaphysics, centered on the concept of ‘form,’ can reasonably be said to begin with the problem of sameness and difference. How can many things, which, in that they are many, are different from each other, nonetheless be the same and so truly bear the same name? How, for example, can many different acts be pious (or impious) (Euthyphro), many different things be beautiful (Greater Hippias), many different states of character be virtues (Meno)? In that, while being different, they are also in some respect the same, there must be something the same about them, or, asSocrates says in the Meno, they must all “have some one same form [ἕν γέ τι εἶδος ταὐτον]” (Men. 72c7). The word here translated, traditionally but inadequately, as ‘form,’ is εἶδος, or, in other similar passages, the related word ἰδέα (e.g., Euth. 5d11, 5e3). As has often been pointed out, these words are related to words for ‘seeing,’ and, less directly, ‘knowing,’ in Greek and other Indo-European languages. Their fundamental meaning is the ‘look’ or ‘appearance’ of something, the way it shows up to the gaze. This is of the utmost importance, for it means that unlike the English word ‘form,’ these words intrinsically and immediately convey a relation to awareness: to say that things have a certain εἶδος is to say something about how they show up or appear to an apprehending consciousness. Many different things “have some one same form” (Men. 72c7) in that they all display the same content to the gaze, and so are truly identified as all pious, all beautiful, or all virtues~ Eric Perl, Thinking Being, 22-23

As to the idea that these exist in a ‘separate realm’ this too is easily misunderstood.

Meaning of Separateness

With our eyes we can see large things, but not largeness itself; healthy things, but not health itself. The latter, in each case, is an idea, an intelligible content, something to be apprehended by thought rather than sense, a ‘look’ not for the eyes but for the mind. This is precisely the point Plato is making when he characterizes forms as the reality of all things. “Have you ever seen any of these with your eyes?—In no way … Or by any other sense, through the body, have you grasped them? I am speaking about all things such as largeness, health, strength, and, in one word, the reality [οὐσίας] of all other things, what each thing is” (Phd. 65d4–e1). Is there such a thing as health? Of course there is. Can you see it? Of course not. This does not mean that the forms are occult entities floating ‘somewhere else’ in ‘another world,’ a ‘Platonic heaven.’ It simply says that the intelligible identities which are the reality, the whatness, of things are not themselves physical things to be perceived by the senses, but must be grasped by thought. If, taking any of these examples—say, justice, health, or strength—we ask, "How big is it? What color is it? How much does it weigh?"we are obviously asking the wrong kind of question. Forms are ideas, not in the sense of concepts or abstractions, but in that they are realities apprehended by thought rather than by sense. ~ p 28

So what’s the point of all this? The point is that the forms do not exist. They’re beyond existence, in that they don’t come into or go out of existence, as do particular things. And the same can be said for mathematical objects:

Neoplatonic mathematics is governed by a fundamental distinction which is indeed inherent in Greek science in general, but is here most strongly formulated. According to this distinction, one branch of mathematics participates in the contemplation of that which is in no way subject to change, or to becoming and passing away. This branch contemplates that which is always such as it is and which alone is capable of being known: for that which is known in the act of knowing, being a communicable and teachable possession, must be something that is once and for all fixed ~ Jacob Klein, Greek Mathematical Thought and the Origin of Algebra.

Of course, all of this was at the very basis of Western science - at least before Galileo. However modern science ‘flattens ontology’. Physicalism in particular, has no conceptual space for the transcendent. We have a ‘flat ontology’ within which existence has a single meaning: something either exists or it doesn’t.

Whereas in the Platonic view, the faculty of reason (nous) is able to ‘peer into the realms of the possible but not-yet-real’ from which it can retrieve discoveries and inventions — such as the device you’re reading this on.

Modern culture has tended to keep the elements of Platonism that are useful for science and engineering — ‘book of nature written in mathematics’ — but jettison the all-important qualitative and ethical aspects which (perhaps unfortunately) had been incorporated into religion.

And here we are.

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I am in agreement about math being real but not being existent, in that mathematical constructs do not live in some cosmic mathland with, as one person I know put it, pi eating smaller irrational numbers.

Furthermore, I am of the view that to actually instantiate a mathematical concept, it needs to be embedded in the physical substrate somehow. E.g. the mathematical concept known as Doom needs a computing device on which to execute, even if John Carmack et al, well, discovered it.

I agree with the mathematical Platonists that math is discovered rather than invented, as the universe did not suddenly start behaving in such a fashion as to obey math-based physical laws when mathematicians came up with the math behind them (which would result in absurd conclusions such as that general relativity only applied starting when Einstein ‘invented’ it).

The only question then is how does one unify mathematical Platonism with the view that math is not merely real but rather is privileged, as Platonism implies that everything physical is an approximation of an ideal Form whereas the position that math is privileged implies that math comes before the physical rather than that the physical merely approximates it.

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I think you have it backwards. Realms far from human experience can be, must be, handled more abstractly than those close to us in scale. At those distances, galaxies and electrons can both be conceived of and studied as points in an abstract space. This is when mathematics works best. Here on Earth, in my living room, I have to worry about the details. I can generally not separate any particular event or phenomenon from its environment.

I think this is a non-sequitur at best. Although I think it is outside the scope of this discussion, scientific reductionism is a contentious position, one which I reject.

Yes, and that’s the whole point.

Previously you wrote “…mathematical constructs are not merely human creations or approximations of an underlying reality, but rather that the underlying reality is implemented in terms of math.” This implies physical laws are the same thing as underlying reality, which I don’t accept. If physical laws are just as much human creations as math is, which I believe, then your argument is circular.

This seems like a stretch of the idea that math is fundamental—that I can know and specify why it doesn’t work in certain situations.

And as I noted, I see this as an argument against your position, not for it.

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People came up with the math in many cases before it was applied to electrons and galaxies, so that the math happens to fit electrons and galaxies so well must then be a very happy coincidence if math were not privileged in the universe.

Note, however, that a viewpoint that readily recognizes emergent behavior in the world is not fundamentally incompatible with a viewpoint that sees the world as being defined in terms of physical laws at the lowest and highest levels.

Just because an abstraction may not be entirely applicable at one level does not mean that abstraction must be ruled out in general.

I agree with the mathematical Platonists that math is discovered rather than invented, and likewise I would say that physical laws are discovered rather than invented.

What I mean is that just because some things, such as the collapse of a wave function, are not deterministic does not mean that a mathematical view of the universe is not applicable, especially since a mathematical view of the universe need not require determinism.

The reason for this, as I see it, is that the further you get away from the most basic physical reality, the more abstraction you introduce, the further you get away from the math underlying everything.

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Certainly — but then there’s the entire realm of pure mathematics, which is not ‘instantiated’ in anything. Pure mathematics describes a real domain far beyond what can be represented instantiated in physical form.

Isn’t it a question of temporal vesus ontological priority? Naturalism tends to see everything in terms of temporal succession unfolding through evolution. But insight into mathematical principles provides access to what is ontologically prior to temporal succession.

But surely the discoveries of quantum physics have undermined any simple concept of physicalism (unless you want to argue that fields are physical). I don’t think any of the founders of quantum physics endorsed physicalism. Werner Heisenberg had this to say:

The Debate between Plato and Democritus

… the inherent difficulties of the materialist theory of the atom, which had become apparent even in the ancient discussions about smallest particles, have also appeared very clearly in the development of physics during the present century.

This difficulty relates to the question whether the smallest units are ordinary physical objects, whether they exist in the same way as stones or flowers. Here, the development of quantum theory some forty years ago has created a complete change in the situation. The mathematically formulated laws of quantum theory show clearly that our ordinary intuitive concepts cannot be unambiguously applied to the smallest particles. All the words or concepts we use to describe ordinary physical objects, such as position, velocity, color, size, and so on, become indefinite and problematic if we try to use them of elementary particles. I cannot enter here into the details of this problem, which has been discussed so frequently in recent years. But it is important to realize that, while the behavior of the smallest particles cannot be unambiguously described in ordinary language, the language of mathematics is still adequate for a clear-cut account of what is going on.

During the coming years, the high-energy accelerators will bring to light many further interesting details about the behavior of elementary particles. But I am inclined to think that the answer just considered to the old philosophical problems will turn out to be final. If this is so, does this answer confirm the views of Democritus or Plato?

I think that on this point modern physics has definitely decided for Plato. For the smallest units of matter are, in fact, not physical objects in the ordinary sense of the word; they are forms, structures or — in Plato’s sense — Ideas, which can be unambiguously spoken of only in the language of mathematics.

Notice ‘whether the smallest units are ordinary physical objects, whether they exist in the same way as stones or flowers.’ This is similar to what I’m arguing about the sense in which numbers are real. They too don’t exist as do ‘stones or flowers’, hence the many idealist readings of quantum physics.

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Consider, however, the mathematics of WiFi, which you’re presumably utilising to conduct this conversation.

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This has been a good discussion, and really useful for me. As I noted at the beginning, this is not something that I’ve worked through satisfactorily by my own standards. That being said, I feel like I laid my position out and I think you’ve had a chance to do the same. I don’t think we’re going to resolve this any more clearly than we have, so I’m going to leave it there.

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I’m not sure how this is relevant to this discussion. I don’t know enough about these subjects to comment intelligently.

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I was unclear about what I meant by ‘instantiated’ — what I mean is that for a mathematical concept to be contemplated in some fashion it needs some kind of physical substrate, as mathematical concepts have no ability of thinking for themselves.

For instance, one can ‘instantiate’ a very large number, a number so big that it is larger than the Eddington number by many orders of magnitude, by thinking about it in terms of Knuth’s up-arrow notation or writing that up-arrow notation on a whiteboard or writing a computer program that operates in terms of symbolic math that can symbolically manipulate up-arrow notation.

Here I meant ontological priority and not temporal priority.

This is why I stated that if one is to try to in some sense preserve physicalist monism, one needs to modify it by making math ontologically precede matter.

This is consistent with a view that is monist but in which math precedes matter rather than matter preceding everything else.

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@Wayfarer’s point is that WiFi, a technology that very many of us use on a daily basis, is fundamentally tied to math in an inextricable fashion, ranging from the math behind the physics of radio waves to the nature of Fast Fourier Transforms (FFT’s) and their inverses. Just because one does not personally see how math has a basic impact in one’s daily life does not mean that it does not have an impact.

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Very good - but how to reconcile that with physicalism? How can the ‘one substance’ be thought of as physical if that is so? How could it be ‘modified’?

Notice that Heisenberg posed that argument in the context of ‘Democritus versus Plato’, where Democritus is the representative materialist (‘all that exists are atoms and the void’) and Plato the idealist, declaring that he believes Plato has been vindicated by quantum physics. (It might also be acknowledged that Heisenberg was an avowed platonist, who was known for carrying a copy of the Timeaus whilst at university.)

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The fact that something has an impact on my life is a completely different subject than whether it is an example of the fundamental reality of mathematics.

As I noted, we’ve reached the limits of my ability to comment with any credibility.

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In this kind of monist scheme, math would form the basis of reality, and the matter forming the physical world would be based upon math, so that the physical world consisting of particles, waves, and fields while existing and being real would not be the most fundamental layer of reality.

Of course, as I have mentioned elsewhere, we have no way of knowing what is truly the most fundamental layer of reality, given my example of the ‘cosmic simulation’ and how even a ‘cosmic simulation’ could itself be a simulation, with the potential for infinite regress.

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My only reaction to this was ‘Well duh…’
It seems that obvious. But there are problems that need resolution. If you think this universe is the real one, why did mathematics choose this particular one?

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It might seem obvious, but I have seen people argue physicalist monism where math is just a pragmatic human invention with no fundamental roots in reality quite earnestly.

As for why this universe is The universe, various arguments have been made ranging from the multiverse combined with the anthropic principle (i.e. any apparent ‘fine-tuning’ is due to fundamental constants being such as to support large multicellular life are required for us to be here to observe them) to intelligent design (e.g. the ‘cosmic programmer’ deliberately choosing constants because they had interesting consequences). Fundamentally, though, we have no way of knowing.

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So really you’re articulating a kind of impasse here. On the hand, mathematics seems more fundamental than the physical entities that supervene on it. But on the other, we have no way of knowing what that fundamental reality actually is, so it might just as well be a simulation.

And I say this is a consequence of those same historical and cultural factors which you declared irrelevant in another discussion. It’s because modern culture started on the basis of the Cartesian dualism of the mind-body division (the ‘substance dualism’ you refer to in the OP). But this dualism was not a viable hypothetical framework because it seemed to sever the link between the two domains and reduce the mind to the status of the ‘ghost in the machine’. So this tends to result in rejecting the idea of ‘res cogitans’ altogether and to try to derive everything from 'res extensia - which, as your analysis shows, results in a kind of radical scepticism and the inability to conceive of anything as ultimately real.

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Sure. I’d even say the majority might categorize it that way. Funny that an alien is likely to ‘invent’ the same math. 2+2=4 everywhere, not just for humans.

Cosmic programmer and ID are the same thing: Just pushing off a problem by positing a bigger problem.
Multiverse and (weak) anthropic, yes, but how multi? Humans (you in particular) appear in several mathematical structure, and maybe most of them do not have ordered laws where you come into being naturally instead of by random chance. This is a real problem with the suggestion that ‘all mathematical structures are treated equally’.