We become aware of our own forms of life because we can compare ourselves to people in other cultures or other eras. Taking that idea deeper isn't uncontroversial, though, is it? Aren't we just speculating that there could be sentient beings who see a radically different world from our own? — frank

Much hangs on what "radically different" means. It could mean that two existing forms of life are incommensurate in such a way that mutual understanding is impossible in principle. Or a form of life could be radically different than our own in the mundane sense that it is difficult for us to fathom prior to having gained some acquaintance with it. Michael Thompson has argued that there is a multiplicity of practical forms of life, while Sebastian Rödl has argued that there is only one. Donald Davidson, in his paper On the Very Idea of a Conceptual Scheme, also argued that the very idea of a multiplicity of mutually incommensurable practical forms of life (or conceptual schemes) is incoherent. That's an issue that Rovelli doesn't contend with. It points to the possibility of a middle term between viewing mathematical theories in a way that make them inherit the contingency of the forms of life which they speak to, on the one hand, and viewing them to be universal in a way that makes them independent of (or unsoiled by) any embodied and situated life form whatsoever, on the other hand.

I've suggested that Kant, and neo-Kantians such as Sellars, are gesturing towards the right kind of necessity with the concept of synthetic a priori propositons — Pierre-Normand

I'd be really interested in any more you have to say about that.

I'd be really interested in any more you have to say about that. — frank

If you don't mind me referring you back to old posts of mine, I've sketched my understanding of the significance of synthetic a priori propositions, here and there, with reference to John Haugeland's neo-Kantian (and Sellars inspired) view of the constitution of the necessary standards (such as mathematical rules, social practices, and/or laws of nature) that make possible objective empirical judgments.

Haugeland's view of constitution, which he further elaborates in his paper Truth and Rule Following, is intermediate between the idea of (contingent) invention of rules and discovery (of 'intelligible' or 'independent' laws).

Here's a bad way to argue:

We're trying to decide between two positions, A and B.

You present an argument that A is true, because of fact X.

Your interlocutor points out that nothing about the truth of X entails A, and so this is a bad argument: the conclusion has nothing to do with the premises.

You respond by saying, well of course it's still possible that B given X, if you take for granted that B!

This is not an effective argumentation strategy.

The fact is that whether or not mathematical objects are interesting to creatures that study or make use of them has nothing to do prima facie with whether there are not such objects. This is, as it stands, a terrible argument. If you want to try to make it better by trying to provide some bridge principles (why should we think that the existence of mathematical objects is contingent on, or interestingly tracked by, the interests of creatures? We're not inclined to think such a thing for any non-human creatures: things are still countable by number, even though there are no creatures who can count!), that's fine, but you can't simply double down.

In other words, we are looking for some reason to accept a premise like:

"A mathematical object exists only if it is interesting to some creature."

This is on its face an absurd claim. Can you give us a reason to believe it?

Thank you so much.

The argument doesn't stand or fall on the question of interest, as I pointed out in the posts on the page previous to this. So I dunno seems a like a bad way to argue is to not read.

I thought you'd be interested on Terence Tao's thoughts on the development of mathematical skill. He has three distinct stages of competence:

(1) Pre-rigorous; like manipulating apples on a table, 2 apples plus 3 apples is... count them.... yes! 5 apples. Or later: 'the derivative of a function is the slope at that point... look if we take the derivative formula for x^2 - yes class that's 2x - and line it up with the point (1,1) on the line, yes! you can see the slope is the same as 2x'.

A pre-rigorous stage mathematician has enough of a feel for a topic to perform basic computations; they will form non-systematic insights and heuristics which they cannot easily translate into formal mathematics. They can follow handle-crank rules without any intuition as to why it works.

A good example here is, probably what most people still have, the ability to times 47 by 12 through the algorithm which looks like:

47
12
-----

and by carrying tens and so on. Most of us could do this with some degree of effort, but god knows precisely how it works. It usually occurs while young and inexperienced with a topic; spanning pre-university and some undergrad education.

(2) Rigorous: like learning how to prove 1+1=2 through a set theory or the Peano Axioms, or how to define fractions and real numbers consistently; epsilon-delta and epsilon-N 'turn the handle' proofs for convergence and continuity. Manually computing derivatives from their definition. This usually occurs mid-undergrad until the masters thesis on a topic, or advanced modules on topics.

A rigorous stage mathematician largely learns definitions and computes proofs and more advanced calculations by reducing them to simpler ones.

Developing the example; a rigorous stage mathematician would be able to prove why the above algorithm for computation works: you can do this by formally summing the stages:
2*7=10+4, remove tens to obtain the units digit, 4, store 10 to the tens computation (total sum 4)
2*40=80, now 90 while adding the 10 (total sum 94)
end stage 1
0*47=0
10*47=470, add previous total sum yielding 564
end
2*7+2*40+10*47=2*(47)+10*47=12*47

(3) Post rigorous: a post-rigorous mathematician can reason intuitively and through analogy about mathematical structures, their intuitions and analogies usually can be translated into proofs, or they will have the ability to find counter examples. The post rigorous stage can be obtained by mathematicians by updating intuitions formed in the pre-rigorous stage to respect the formalisms learned in the rigorous stage. This obtains after advanced modules or long theses are completed, typically the sphere of research or practicing mathematicians in their required competences.

A post rigorous modification of the above algorithm works to compute the product in binary:


I winged this off the intuition that 'carry the ten' was the same as 'carry the base' and filled out the details later.

To be sure, post-rigour is topic specific; while I'm sufficiently familiar with positional notation and base changes to intuit quite a lot about them, I'm nowhere near as familiar with formal proofs in ZFC, so I couldn't wing deductions in ZFC by 'getting the right idea' first then 'working out the details'.

The motivation for bringing this up being that how the conceptual links work between synthetic a-priori propositions, or synthetic a-priori ideas/intuitions, are competence dependent and topic specific. Which isn't to say that the fact that the algorithm above generalises to any base system is dependent upon my competence, but the expression of that fact depends entirely on the 'right links' and 'right intuitions' being in place in their expresser. So there's a historical component to it; there's no way I'd've been able to wing the above generalisation without floating point arithmetic, hex and binary manipulations being part of computing courses. Having the 'right links' and 'right intuitions' is as much a function of the intellectual milieu as creative competence.

Which posts are you referring to?

Would you rather people address the argument in this post:
https://thephilosophyforum.com/discussion/comment/220479

Than the one in the OP (they are not the same)?

Strictly speaking, according to the argument, it's invalidated because it can't account for why math is the way it is. That an account of mathematics should explain why math is the way it is, is a premise (which MP fails to satisfy). If you don't like the premise, you can reject it and offer another in its place, no? — Πετροκότσυφας

MP posits the existence of a realm of independent, abstract, mathematical objects. Why should this ontology be required to "explain why [our] math is the way it is"?

Idc which post you address; just asking that you do a better job of it.

The geometry on a blank paper and the geometry on a sphere are different, but their existence doesn't make one or the other illogical. — ssu

It's not that one or the other principle is illogical, because it requires two opposing principles for there to be contradiction, and the contradiction is what is illogical. That the object, space, is described in these two contradictory ways is what is illogical.

That argument is wrong, but it doesn't make either geometry illogical. Especially in set theory you can choose your axioms and have different kinds of set theories with different answers, but that in my mind don't make them illogical. — ssu

What is the case, is that this subject, mathematics, which allows that the objects which it deals with are described in contradictory ways, is illogical. That one geometry is consistent within its own system, yet inconsistent with another geometry, or one kind of set theory is consistent within its own system, yet in inconsistent with another set theory, doesn't make any of them, in themselves, illogical. But this is not the issue. The issue is that the discipline of mathematics, which allows within it, such inconsistencies, is illogical. It's not that this or that branch of mathematics is illogical because this branch is inconsistent with that branch, but that mathematics itself is illogical for allowing such inconsistencies within the discipline.

Premises (axioms) can make the math to seem contradictory, but can be totally logical. Only if you prove that something that we call an axiom is actually false, then is the statement simply wrong. — ssu

Opposing axioms cannot both be true. Therefore, one or the other, or both must be false. It is illogical to hold opposing axioms, as both true, and this is what mathematics does.

Thank you for so convincingly demonstrating the patent absurdity of nominalism. — aletheist

Yes, the argument is very simple and clear, isn't it? So much so that you have no counterargument except, "that's absurd". I thought you said that the argument is "self-refuting". Is that how it is refuted, by you saying it's absurd?

But you said the argument had nothing to do with interest, and the argument in the first post very explicitly does. In order to address the issue, you need to be clear about what you're actually claiming.

I think both arguments are very bad, but the response will change depending on which is addressed.

The argument relies on the acknowledgement that the manner in which we sort out the wheat (fruitful mathematical theories) from the chaff (unprincipled and uninteresting sets of axioms) reflects contingent features of our specific form of life. — Pierre-Normand

Good post. I agree it is about finding the (pragmatist) middle path.

So in my simple-minded way, the debate is about the interaction between the formal and the material in a hylomorphic conception of nature. Maths stands for a notion of all the forms that are possible. And as such, it is unconstrained by material considerations. Practical considerations of energy and matter - whatever it is that instantiates a form as something actually physical - are left out of the story.

And that disconnection is where all the confusion arises. It in fact is the same confusion that leads to mind~body dualism. It is only once psychology is understood as something embodied and function serving that we can see why physics acts to produce the kind of "information processing" nervous system that we see.

So the maths that matters is the maths of purpose-serving structures. They do reflect the features of a form of life. But the features need not be contingent, nor the way of life just ours. Instead, we could find our centre ground in the standard structuralist claim that there are generically necessary "forms of life". For actualised substance to be the case, there has to be a systematic organisation - that interaction between downward constraining forms and upward constructing material degrees of freedom.

So tracking the genericity of structures leads you towards a thermodynamic, probabilistic, and indeed semiotic, metaphysics. Life is certainly a form of thermodynamics - a semiotic elaboration on a dissipative structure. And even the Cosmos is a "form of life" in being a dissipative structure at a universal level.

In the grand scheme then, we would seek to unite the formal and the material in the substantial. That hylomorphism must be the thread that connects existence at its every level, its every stage of complexity. So substantial reality can be said to have a general interest in the formal structures that work - the structures that can harness material flows or entropy to achieve the goal of wresting stablility from impermanence. A "way of life" boils down to just that. And the maths that matters most, the maths that has true reality, is that which describes the kinds of forms or structures which subserve that fundamental purpose.

So if the mathematical realm is some vast landscape of possible algorithms, then only a limited number would be highly effective at doing the job of harnessing material instabilities. The "good" maths would be that which has the emergent property of producing finitude or constraint. Junk maths just wanders off forever in open-ended fashion. Good maths - like symmetry maths or fractal maths - speaks of nature because it speaks of self-organising limitation and closure.

Of course, mathematicians are free to explore the "junk". They could just play with open-ended patterns as a matter of contingent human choice. The constructed view of maths could be true as part of the socially-accepted "mathematician's way of life". :)

But still, humans have historically valued the maths that could speak to natural structure. And natural structure is hylomorphic. What matters is how constraints can organise freedoms to produce substantial actuality.

We can kick around Plato as if he were some kind of ancient mystical fool. But from my own readings, what strikes me is how dimly we appreciate the systems-based revolution that was Greek metaphysics - especially in the writings of Anaximander and Aristotle. It was how science was born. And then the systems view got obscured because causal reductionism took over. Nominalism made formal/final cause appear mystic and uncool. And we have been living with that metaphysical confusion ever since.

from my own readings, what strikes me is how dimly we appreciate the systems-based revolution that was Greek metaphysics - especially in the writings of Anaximander and Aristotle. It was how science was born. And then the systems view got obscured because causal reductionism took over. Nominalism made formal/final cause appear mystic and uncool. And we have been living with that metaphysical confusion ever since. — apokrisis

:up:

Characterized by forms, reality had an intrinsic intelligibility, not just in each of its parts but as a whole. With forms as causes, there are interconnections between different parts of an intelligible world, indeed there are overlapping matrices of intelligibility in the world, making possible an ascent from the more particular, posterior, and mundane to the more universal, primary, and noble. In short, the appeal to forms or natures does not just help account for the possibility of trustworthy access to facts, it makes possible a notion of wisdom, traditionally conceived as an ordering grasp of reality. — J P Hochschild

I thought you said that the argument is "self-refuting". — Metaphysician Undercover

I said that recognizing some judgments as true and others as false entails that there is a fact of the matter, which is independent of whatever anyone thinks about it; and that any argument to the contrary is self-refuting. Why? Because disputing it requires presupposing it.

Is that how it is refuted, by you saying it's absurd? — Metaphysician Undercover

You seem to be asserting that something is not real unless and until a word for it exists, which is what I find patently absurd. The reality of (what we call) roundness and the world does not depend on the existence of those names. The world was real, and was really round(ish), before humans ever existed.

I said that recognizing some judgments as true and others as false entails that there is a fact of the matter, which is independent of whatever anyone thinks about it; and that any argument to the contrary is self-refuting. — aletheist

I can't see your argument. If some judgements are true, and others false, then truth and falsity is a property of the judgement. Therefore it is impossible that truth and falsity are independent of the judgement.

You seem to be asserting that something is not real unless and until a word for it exists, which is what I find patently absurd. The reality of (what we call) roundness and the world does not depend on the existence of those names. The world was real, and was really round(ish), before humans ever existed. — aletheist

No I am saying that there is no such thing as the thing referred to by a word without the word. How is that absurd? Therefore it is impossible that there was a thing which the word "world" refers to before there was the word "world". To say that there was a reality of roundness before there was a word "round" is what is absurd. What would dictate what roundness is without the word and a corresponding concept? So how could there be a reality of roundness without this?

If some judgements are true, and others false, then truth and falsity is a property of the judgement. Therefore it is impossible that truth and falsity are independent of the judgement. — Metaphysician Undercover

I never suggested otherwise. However, if a particular judgment is true, why is it true? And if a particular judgment is false, why is it false? In both cases, the answer is that there is a fact of the matter, and that fact is independent of whatever anyone thinks about it. A true judgment represents a fact, while a false judgment does not.

No I am saying that there is no such thing as the thing referred to by a word without the word. How is that absurd? — Metaphysician Undercover

Because it entails that the reality of an object somehow depends on the existence of a sign that represents it; but reality is precisely that which is as it is regardless of any representation thereof. In other words, a real thing (or quality or habit) is that thing (or quality or habit) regardless of whether there is any word that refers to it. The thing (or quality or habit) came before the name that some humans arbitrarily invented for it. There is no necessary connection between most words and most things (or qualities or habits), only a convention by which the words refer to the things (or qualities or habits) within a particular language or other system of signs.

What would dictate what roundness is without the word and a corresponding concept? — Metaphysician Undercover

This is exactly backwards. What would prompt the creation of the word "round" if there was nothing already observable for which such a name was needed?

Because that's what the paper answers to, I'd assume. In the referenced work by Penrose, it is stated that: "It may be helpful if I put the case for the actual existence of the Platonic world in a different form. What I mean by this ‘existence’ is really just the objectivity of mathematical truth. Platonic existence, as I see it, refers to the existence of an objective external standard that is not dependent upon our individual opinions nor upon our particular culture".

So, math is the way it is because it can't be otherwise. There's something outside all human practice (a platonic realm where mathematical truths reside, waiting to be discovered, say), which makes math the way it is. The paper tries to refute this. — Πετροκότσυφας

I find the paper's attempted refutation to be unsuccessful.

The author says if M is too large then it is uninteresting. But what the author means to say is that if M is too large then it is mostly uninteresting to us (and perhaps also: at this point in time). Obviously, there are some parts of M that we do find interesting, because "the value is in the selection, not in the totality". The author says that M "contains too much junk", but again, as judged by us (or, alternatively, by the Jovians).

The Jovians might have a different mathematics to us and they might find different parts of M more useful than we do, but presumably we and the Jovians each borrow true theorems and objects from the same universal M. Likewise, we may have discovered how interesting linear algebra is only recently, but it could always have been a part of M, waiting there for us to use it. The same can be said of 2d and 3d (and perhaps 4d) geometry, arithmetic, set theory, logic, category theory, topos theory and whatever mathematics the Jovians might use.

The author says that if M is smaller and interesting then it is not independent. But then neither is it the M of mathematical platonism, I would argue.

The author says that the complete independence of MP "is like the idea of an Entity that created the heavens and the earth, and happens to very much resemble my grandfather."

Here I agree with the author. I think he has as much chance of refuting MP as he would have of refuting the existence of God.

However, if a particular judgment is true, why is it true? And if a particular judgment is false, why is it false? In both cases, the answer is that there is a fact of the matter, and that fact is independent of whatever anyone thinks about it. A true judgment represents a fact, while a false judgment does not. — aletheist

I think you are wrong here aletheist. True judgement implies correspondence. Why is it true? It is true because there is correspondence between the judgement and reality. You have provided no premise whereby you can claim that if there is correspondence, one of the corresponding things is necessarily independent of the other.Therefore your conclusion of an independent fact is illogical, invalid, non sequitur. "True" implies correspondence which implies a relationship, it does not imply independence.

Because it entails that the reality of an object somehow depends on the existence of a sign that represents it; but reality is precisely that which is as it is regardless of any representation thereof. — aletheist

The reality of an object is dependent on the process whereby the object is individuated from its environment the rest of reality. This process is carried out by sentient beings. Unless you can demonstrate how this process of individuation is carried out by something other than sentient beings, or demonstrate how this is false, then you have no argument.

Consider this example. We perceive the earth as an object. But we know that the earth is not an independent object, it is part of the solar system. We individuate it, separate it from its context, and assign to it the status of "an object", when it is just as appropriate to say that it is not an object, but part of an object, the solar system. Its independence as "an object" is artificial, created by us. Its status as an object is simply a product of how we individuate things. There are many such examples in scientific theory, electrons, and protons are said to be objects. Science is full of things which we know to exist only as parts, but we treat them in theory as independent objects.

The thing (or quality or habit) came before the name that some humans arbitrarily invented for it. — aletheist

Whether or not the thing came before the symbol which represents it, is debatable, The thing, the object, is dependent for its existence as an object, on the process of individuation carried out by the sentient being which individuates it, as described above. To determine which "came before", we must consider the temporal nature of this process. The sentient being exists at the present, and assigns the status of "object" to things observed to have temporal extension, things which remain relatively unchanged over a period of time, or demonstrate temporal continuity. Since the being exists only at the present, this, observing temporal continuity, requires memory of what has been in the past. And memory requires that what has been, is represented. Representation requires symbols. So I believe that the logic demonstrates that the symbol is prior in existence, to the object.

What would prompt the creation of the word "round" if there was nothing already observable for which such a name was needed? — aletheist

Imagination! Imagination may not be something real to you, but it is the inspiration behind creation.

What did I insert into the argument that was not there? SX's argument is that MP cannot explain why our mathematics is but "an infinitesimal subset" of M. This again implies our interest. And even if SX's argument doesn't imply this (it does), then the paper of the OP's argument explicitly refers to our interest, anwyay.

I have no motivation to defend MP as I'm not a platonist. I'm just pointing out that the author's supposed refutation fails, or at the very least it fails to convince me. Do you have a knock-down argument to refute the existence of God?

SX's argument is that MP cannot explain why our mathematics is but "an infinitesimal subset" of M. This again implies our interest. — Luke

This is incredibly silly. A failure of X to explain Y cannot entail that Z must in turn be implicated. That's just a basic failure of logical form, let alone content.

If you are referring to my account of your argument, then:

P1. Any account of mathematics would need to explain why mathematics is the way it is.
P2. Mathematical Platonism is the view that there is a world M, that contains all possible mathematical objects and truths.
P3. Mathematics is but "an infinitesimal subset" of any such mathematical reality.
P4. Any account of mathematics would need to explain why P3 is the case, in order to satisfy P1.
P5. Mathematical Platonism has no way to explain why P3 is the case.
C1. Mathematical Platonism cannot satisfy P1.

Ergo, Mathematical Platonism fails to have any explanatory force with respect to mathematics.

If you are referring to whether or not your argument implies our interest, it's about our mathematics. Do you claim that we have no interest in our mathematics, or (i.e.) in our infinitesimal subset of M?

Neither. Read, or stop talking.

I beg your pardon, admin?

I beg your pardon, admin? — Luke

I'm not interested enough to read it either (ha ha). I think we're ultimately limited to phenomenology of such things. I am interested in reading more Haugeland, though, (as soon I discover where his essays are stashed.)

I am interested in reading more Haugeland, though, (as soon I discover where his essays are stashed.) — frank

When we discussed Pattern and Being, it was available online. Truth and Rule Following (which further develops the same ideas) only has been published as the last chapter of Having Thought: Essays in the Metaphysics of Mind, HUP, 2000. I can't legally post pdf documents but I can PM you a treasure map.

When we discussed Pattern and Being, it was available online. Truth and Rule Following (which further develops the same ideas) only has been published as the last chapter of Having Thought: Essays in the Metaphysics of Mind, HUP, 2000. I can't legally post pdf documents but I can PM you a treasure map. — Pierre-Normand

Awesome! Thank you!

I thought you'd be interested on Terence Tao's thoughts on the development of mathematical skill. — fdrake

Thanks very much for that. I think this idea, suitably adapted, might dovetail nicely with Wiggins's account of someone's conception of a concept, construed as the Fregean sense of this concept (while the concept itself still lives at the level of Bedeutung, or of Fregean reference). Tao's idea of the development in skill/understanding lines up with Wiggins's idea of the improvement of a conception that enables, at once, a better grasp of the sense of a concept and an active participation into its constitutive practice.

Wiggins develops this idea most fully in The Sense and Reference of Predicates: A Running Repair to Frege's Doctrine and a Plea for the Copula

Just like the pragmatized (embodied and situated) neo-Kantian account of Sellars, Haugeland and Bitbol, Wiggins's account of the way in which we grasp concepts steers a middle path between anchoring them into merely contingent features of the embodied subject or making them fully 'independent' of us (in the manner of modern Platonism). The precise account of this cognitive anchoring, though, appeals to some features of modality and reference that are indebted to Frege and to Kripke (and Putnam). Those features have been highlighted by Gareth Evans, also, in The Varieties of Reference (in the chapter on proper names, which discusses reference to natural kinds, also). I've written some posts about this many years ago on a Yahoo discussion group. I'll try to locate them.(*)

(*) Here they are: see mainly this post, which was a followup on this one.

I continue to find your argumentation nonsensical, and have decided to stop wasting my time with it. Cheers.

:up: — Wayfarer

(In response to this post by @apokrisis)

Seconded!