Mathematical Platonism and its Critics

[j_j]

@Wayfarer

This is what this means: when you think, you see - mentally see - a form which could not, in principle, be identical with a particular - including a particular neurological element, a circuit, or a state of a circuit, or a synapse, and so on. This is so because the object of thinking is universal, or the mind is operating universally.

Of course this is a traditional way of “seeing” it. But it’s a thesis to be defended, is it not ?

If this is “what we all really do,” then why is the thesis controversial ? If the mind’s thoughts are “universal,” then why do we misunderstand and disagree with one another ?

We might explore understanding “ideality” in terms of the normative. We strive toward a senes of shared “meaning” — a sense of sharing in aspect of the world.

From my POV, your math platonism seems very “local,” based on the positive integers, etc. I’m curious how you might react to the wilder flights of set theory, if you find them intuitive, there for the internal eye of pure intellect.

In other words, I think there’s a tension between the Intuitionism in your view and the Platonism. Brouwer and others objected to the injection of non-intuitive linguistic thinking into a relatively pure numerical intuition.

If I have 100 cardboard boxes, I can say that either all of the boxes are empty or there’s at least one non-empty box. That makes sense. We can check.

But what I have an infinity of cardboard boxes ? Can I say that they are all empty or at least one is not ?

[Sime]

@j_j

Perhaps you overlook the transformation of the hard problem of consciousness into the hard problem of being or the hard problem of the physical. As I see it, we need not start from dualism. I’m not sure where your own starting point may be.

Yes, for the purposes of philosophical clarity it is better to start by avoiding dualism of any sort. This is why I suggest analysing meaning and truth using semiotic analysis, since signs are the basis of an individual’s understanding of his world, and these signs cannot be pigeonholed into the neat physical, psychological, and mathematical semantic categories of our public language.

C.S Peirce was the primary founder of modern symbolic logic, and he attempted to grasp its foundation informally in terms of his triadic system of semiotics, that resonates with very recent formal approaches to the same foundational questions.

[Wayfarer]

@j_j

Of course this is a traditional way of “seeing” it. But it’s a thesis to be defended, is it not ?

That was an excerpt from a Gerson lecture, in which he very briefly paraphrases Aristotle’s argument from D’Anima. Aristotle’s epistemology is participatory — ‘in thinking, the intelligible object or form is present in the intellect, and thinking itself is the identification of the intellect with this intelligible.’ But ‘thinking’ is probably a rather weak translation of what the original term, presumably based on ‘nous’, that seminal Platonic term now translated as intellect.

@j_j

why is the thesis controversial ? If the mind’s thoughts are “universal,” then why do we misunderstand and disagree with one another ?

It is a reference to universals, which are how ideas (eidos) are understood in Aristotle. In hylomorphism, every particular is a composite of matter and form; the form is the universal, the ‘what-it-is-ness’ that confers an object’s identity.

This has really nothing whatever to do with ‘infinite cardboard boxes’. Or C S Peirce, for that matter.

[j_j]

I’m a fan of Peirce, though I don’t claim to have digested his semiotics. Have you looked into Bakhtin ? A recent discovery for me, but I like him for “seeing” the concrete speech act.

But the monologic utterance is, after all, already an abstraction, though, to be sure, an abstraction of a ‘natural’ kind. Any monologic utterance, the written monument included, is an inseverable element of verbal communication. Any utterance — the finished, written utterance not excepted — makes response to something and is calculated to be responded to in turn. It is but one link in a continuous chain of speech performances. Each monument carries on the work of its predecessors, polemicizing with them, expecting active, responsive understanding, and anticipating such understanding in return. Each monument in actuality is an integral part of science, literature, or political life. The monument, as any other monologic utterance, is set toward being perceived in the context of current scientific life or current literary affairs, i.e., it is perceived in the generative process of that particular ideological domain of which it is an integral part … The isolated, finished, monologic utterance, divorced from its verbal and actual context and standing open not to any possible sort of active response but to passive understanding on the part of a philologist — that is the ultimate ‘donnee’ and the starting point of linguistic thought. …

The problem of meaning is one of the most difficult problems of linguistics. Efforts toward solving this problem have revealed the one-sided monologism of linguistic science in particularly strong relief. The theory of passive understanding precludes any possibility of engaging the most fundamental and crucial features of meaning in language …
..
To understand another person’s utterance means to orient oneself with respect to it, to find the proper place for it in the corresponding context. For each word of the utterance that we are in process of understanding, we, as it were, lay down a set of our own answering words. The greater their number and weight, the deeper and more substantial our understanding will be. Thus each of the distinguishable significative elements of an utterance and the entire utterance as a whole entity are translated in our minds into another, active and responsive, context. Any true understanding is dialogic in nature.
…
Understanding is to utterance as one line of a dialogue is to the next. Understanding strives to match the speaker’s word with a counter word. Only in understanding a word in a foreign tongue is the attempt made to match it with the ‘same’ word in one’s own language. Therefore, there is no reason for saying that meaning belongs to a word as such. In essence, meaning belongs to a word in its position between speakers; that is, meaning is realized only in the process of active, responsive understanding.
…
Meaning does not reside in the word or in the soul of the speaker or in the soul of the listener. Meaning is the effect of interaction between speaker and listener produced via the material of a particular sound complex. It is like an electric spark that occurs only when two different terminals are hooked together. Those who ignore theme (which is accessible only to active, responsive understanding) and who, in attempting to define the meaning of a word, approach its lower, stable, self-identical limit, want, in effect, to turn on a light bulb after having switched off the current. Only the current of verbal intercourse endows a word with the light of meaning.

I think the triadic approach includes this recognition ?

Perhaps you’d like to contribute to the thread about signs and meaning.

[j_j]

@Wayfarer

This has really nothing whatever to do with ‘infinite cardboard boxes’.

The cardboard box issue connects directly to platonism about the natural numbers. The intuitionists answer the question one way, and platonists another.

Peirce looks relevant to me, because a sophisticated challenging of platonism will not just be boring culture-bias. It is likely to be semantic. As in what do these platonists even mean ? One way to make their claims concrete is for them to declare which mathematical statements, if any, are “true” or “false” apart from our ability to determine which.

Some views in the philosophy of mathematics are object realist without being platonist. One example are traditional intuitionist views, which affirm the existence of mathematical objects but maintain that these objects depend on or are constituted by mathematicians and their activities.
…
Truth-value realism is the view that every well-formed mathematical statement has a unique and objective truth-value that is independent of whether it can be known by us and whether it follows logically from our current mathematical theories. The view also holds that most mathematical statements that are deemed to be true are in fact true. So truth-value realism is clearly a metaphysical view. But unlike platonism it is not an ontological view. For although truth-value realism claims that mathematical statements have unique and objective truth-values, it is not committed to the distinctively platonist idea that these truth-values are to be explained in terms of an ontology of mathematical objects.

Mathematical platonism clearly motivates truth-value realism by providing an account of how mathematical statements get their truth-values.
source

@Wayfarer

It is a reference to universals, which are how ideas (eidos) are understood in Aristotle. In hylomorphism, every particular is a composite of matter and form; the form is the universal, the ‘what-it-is-ness’ that confers an object’s identity.

Of course. But I’m asking for a defense of “universals.” And I don’t do so as a “materialist.” I think mathematics and language generally are “meaningful.” The issue is getting a grip on this “significance.” In this thread, it’s about mathematical meaning and truth, is it not ?

I have no objection to Aristotle or Plato. I like Plato. But I am more of an intuitionist than a platonist, because intuitionism insists on meaning without denying time.

[Wayfarer]

@j_j

If the mind’s thoughts are “universal,” then why do we misunderstand and disagree with one another ?

I have something further to add. If ‘the mind’s thoughts’ are of mathematics, then there is no scope for disagreement. Mathematical truths are not ours to choose. Of course, the same can no longer be said of the other ideals of platonism — beauty, justice — in our hyperpluralistic culture. But the underlying idea that universals are the ground of agreement between different individuals; that is part of what makes them universal!

Also I mispoke when I said C S Peirce was irrelevant. There’s a point on which he is relevant, in that he maintained a realist view of universals — ‘realism’ meaning ‘scholastic realism’, accepting the reality of universals.

@j_j

I’m asking for a defense of “universals.”

It’s a very big topic, a proper defense would require a lot of text. But in respect of the question posed in this particular OP, it’s not difficult to discern the underlying fault lines. It is pretty well precisely the rejection of the idea of universals which underlies the general hostility towards platonism.

[Wayfarer]

@j_j

One example are traditional intuitionist views, which affirm the existence of mathematical objects but maintain that these objects depend on or are constituted by mathematicians and their activities.

Which has serious difficulties explaining ‘the unreasonable effectiveness’ argument: why it is that mathematical discoveries made in respect of some subject, suddenly can be applied to subjects with no apparent connection. And why it is that mathematical predictions can be used to discern attributes of nature which could otherwise never have been discovered.

[j_j]

In some sense, I’m a realist about universals. But I’m not a dualist. The hard part is digging out that sense. Communication is difficult !

Hence the Bakhtin quote. Philosophers tend to assume that they are intending the same “thing” by the same words. So sharing in “meaning” — in a sense of the situation ---- is ideal. As in something we chase.

One critique of one conception of universals is that they are “out there” in a frozen state. To me the positive numbers are especially “sturdy” in this regard. So that would be the “easiest” or most immediately plausible restriction of platonism. Even here, the intuitionist reject the LEM in the case of infinite sets as insufficiently meaningful. Yet the intuitionists are no less invested in a metaphor of “inner seeing.”

[j_j]

@Wayfarer

Which has serious difficulties explaining ‘the unreasonable effectiveness’ argument: why it is that mathematical discoveries made in respect of some subject, suddenly can be applied to subjects with no apparent connection. And why it is that mathematical predictions can be used to discern attributes of nature which could otherwise never have been discovered.

That’s a rich issue. One angle of approach is to simply consider how powerful it is to count carefully. More importantly perhaps, the system of fractions allows for arbitrary precision measurement.

But why are we so lucky to find inverse square laws ? It makes sense that simpler laws would be discovered/invented first.

Hamming might remind us that science has explained very little. But what it has explained has changed our lives.

[Wayfarer]

The question I posed at the outset, is in what sense are numbers (being a type of universal) real? In the mind, or in the world? Because many will say that those are the only options - they’re either objective, inherent in nature, or subjective, the product of the mind. Whereas I say their reality inheres in the relation of mind and world. Hence, my saying that Aristotle’s epistemology is poarticipatory - the mind ‘becomes’ the universal. That is made explicit in scholastic philosophy.

if the proper knowledge of the senses is of accidents, through forms that are individualized, the proper knowledge of intellect is of essences, through forms that are universalized. Intellectual knowledge is analogous to sense knowledge inasmuch as it demands the reception of the form of the thing which is known. But it differs from sense knowledge so far forth as it consists in the apprehension of things, not in their individuality, but in their universality.

Do you think I like advocating Catholic philosophy? I’m not even Catholic! But I find this reasoning compelling.

[j_j]

For me the mind becomes the universal to a maximum intensity. Consciousness as the being or presence of world-from-POV means that , as consciousness, I am the being of the tiger. This tiger “shows itself” in “fugitive flashes” ( sense-perception, etc.) But I don’t see a random blur of orange and black. I see the unified, enduring tiger itself.

This unification of “aspects” over time, space, and sites of consciousness is ideally universal. I tried to point out, and I think @Sime made a similar point, that the most “meaningless” symbols of the mathematical formalist already involve an “ideality.” Curry, a famous formalist, was well aware of this. For instance, some formalist metamathematics uses |,||,|||, … as formal numerals. But obviously it’s not the inkshape on a particular piece of paper that’s intended. It’s already the “idea” of such a vertical stroke. As Derrida emphasizes, ideality and repeatability are profoundly connected. But he would say, and I agree, that there is no perfect repetition. None of the actual marks on paper are exactly the same, but they are treated as the “same abstract mark” within a formal system.

[Joshs]

@j_j

Hamming might remind us that science has explained very little. But what it has explained has changed our lives

Has what science explained changed our lives in spite or apart from philosophy and the arts, or in inseparable enmeshment with them? If the latter, then isn’t it the philosophy underlying the science
which gives it
its power?

[Wayfarer]

@j_j

Hamming might remind us that science has explained very little. But what it has explained has changed our lives.

He wrote:

Science in fact answers comparatively few problems. We have the illusion that science has answers to most of our questions, but this is not so. From the earliest of times man must have pondered over what Truth, Beauty, and Justice are. But so far as I can see science has contributed nothing to the answers, nor does it seem to me that science will do much in the near future. So long as we use a mathematics in which the whole is the sum of the parts we are not likely to have mathematics as a major tool in examining these famous three questions.

[j_j]

@Joshs

Has what science explained changed our lives in spite or apart from philosophy and the arts, or in inseparable enmeshment with them? If the latter, then isn’t it the philosophy underlying the science
which gives it
its power?

Sure. There is a philosophy enacted in science, you might say. Not knowledge is power but power is knowledge. This I claim is the dominant human epistemology. But of course “power” is ambiguous here. I use it to point at the qualitative “evidence” — us getting what we want reliably — the secures the prestige of science via its essence, technology.

[Count_Timothy_von_Icarus]

I seem to have lost the post I had written, but there is an interesting Scholastic/Islamic argument for the priority of universals to magnitude and multitude (number/ratio) that runs through parallel phenomenological and metaphysical tracks. I figured the phenomenology one might interest you more.

Going back to Aristotle, there is the idea that recognizing wholes (units) must be prior to recognizing number. For, if one cannot identify any whole (e.g., a cat), it seems impossible for one to recognize multiple wholes, i.e., multitude. Likewise, without some unit (and so some measure), one could hardly speak of half, a quarter, and so on.

So the whole is prior in cognition, and unity to unit itself. That’s the order of knowing, and the argument from the order of being is a bit more involved.

With Aquinas, he sees dimensive/bulk quantity (number) as being dependent on more basic qualities. If nothing was anything in particular, there would be no way for it to be one, or two, or half, etc. There is no “half as hot” without heat, no “three men” without men. So the primitive quantity he has in mind, called virtual quantity (relating to power), is not numerical, but has to do with qualitative intensity.

Anyhow, this is interesting in a modern context because it puts phenomenal whatness, quality, prior to number, whereas the flip that occurs with materialism tends to have number as a sort of bare actuality that is prior to all phenomenal whatness or noetic content (which increasingly become extrinsic impositions of the mind in materialist philosophy). This means that the older sort of mathematical platonism tends to have number as a sort of intermediary in a hierarchical chain of principles and causes, whereas modern mathematical platonism tends towards disconnected, bare “abstract objects.” I would say the two are actually quite different.

There are obviously some similarities to later phenomenology here too.

[Joshs]

@j_j

@Joshs

Has what science explained changed our lives in spite or apart from philosophy and the arts, or in inseparable enmeshment with them? If the latter, then isn’t it the philosophy underlying the science
which gives it
its power?

Sure. There is a philosophy enacted in science, you might say. Not knowledge is power but power is knowledge. This I claim is the dominant human epistemology. But of course “power” is ambiguous here. I use it to point at the qualitative “evidence” — us getting what we want reliably — the secures the prestige of science via its essence, technology

Let me put this differently. Heidegger organizes the history of Westen culture into epoches, Foucault talks of epistemes, and Wittgenstein of forms of life. At least for Heidegger and Foucault, cultural eras weave together all aspects of culture on the basis of shared metaphysical presuppositions. Accordingly, the methods of science have no advantage over the other cultural forms of expression in terms of access to ‘truth’. Science is simply a conventionalized, applied mode of philosophical inquiry, and to the extent that we can define some notion of progress associated with science and technology, that same notion applies equally to philosophy.

[Wayfarer]

About a decade ago, I was looking through the philosophy shelves in Abbey’s Bookshop in Sydney, when I overheard a conversation between someone who seemed to be a philosophy professor and what I presumed was one of his students, who was deliberating as to whether to do postgrad work in philosophy. This professor had a very plummy Australian English diction and sounded pretentiously erudite. He had plenty to say. A couple of snippets have stayed with me. One was that in philosophy, ‘the Greeks, the Medievals, the Germans — that’s all you need to know. The rest is rubbish!’ (although I had the feeling it was slightly tongue-in-cheek.) Another was— and this was in all seriousness— we longer have a credible cosmology. I was at bit puzzled by that, but later I encountered Alexander Koyré, From the Closed World to the Infinite Universe, and I could see what he meant.

Compare:

@Hamming

From the earliest of times man must have pondered over what Truth, Beauty, and Justice are. But so far as I can see science has contributed nothing to the answers, nor does it seem to me that science will do much in the near future.

all of which, I feel, is apposite in any discussion of Plato.

[j_j]

Great post, and I’d enjoy whatever elaboration you feel like adding to it.

@Count_Timothy_von_Icarus

Going back to Aristotle, there is the idea that recognizing wholes (units) must be prior to recognizing number. For, if one cannot identify any whole (e.g., a cat), it seems impossible for one to recognize multiple wholes, i.e., multitude.

This makes sense to me. The object as such is a unity. On the other hand, there is some thing that is “sensory” and “emotional” about world/experience that “exceeds” conceptualization. Of course attaching a conceptualized quality like redness to an object is to include that redness in a unity, something like substance. To me it looks like substance is “logical.”

@Count_Timothy_von_Icarus

this is interesting in a modern context because it puts phenomenal whatness, quality, prior to number, whereas the flip that occurs with materialism tends to have number as a sort of bare actuality that is prior to all phenomenal whatness or noetic content

Right. This is basically what I mean when I say that a certain flavor of physicalism is tacitly disavowed idealism. The “true world” is the “meaning” of various mathematical models.But there’s a tension here, because the thrust of this kind of physicalism is toward something radically external to all subjects.

[j_j]

@Wayfarer

all of which, I feel, is apposite in any discussion of Plato.

I agree. So I’d maybe invoke the normative in general here. We care about this stuff. We seek for something valid beyond just our hunches and preferences.

[j_j]

I find all of that quite reasonable. So I don’t disagree.

And yet I want to emphasize the glamor and prestige of science even among those who hate math and hate atheism. Science is technology that works whether or not you believe in it ( expect it to.) And whether or not you understand it. This is of course not a sublime definition of science. It’s strategically blunt and crude.

Cultural critics don’t usually ignore the brutal persuasiveness of technology. They can’t afford to do so.

I’m pointing away from sublime assimilations of science by a generalized concept of philosophy, even if such an assimilation is validly asserted in some sense. The man on the street doesn’t have time for such intricate longwinded stuff. “Show me the money/power.”

A wicked and adulterous generation seeketh after a sign