Mathematical Platonism

[Pollinosis]

Mathematical Platonism is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices.
Do you agree with this view?

[Pollinosis]

Do those of you who selected 'no' subscribe to some form of mathematical nominalism?

[Ahknaton]

I believe in Mathematical Platonism. There are a bunch of arguments that can be dredged up to support the Platonist point of view, but to be honest they are retroactive justifications for what is already a firmly held belief. For me the tipping point came when I was about 14 and had programmed a simple Mandelbrot Set program into AmigaBasic. The experience that I was actually exploring something with its own existence that had been sitting there waiting to be discovered for eternity before Benoit Mandelbrot found it (as opposed to telling my computer to render something that had been defined into some kind of contingent existence by him) was and is too hard to shake.

[Pollinosis]

I have to say, the idea that mathematical truths would somehow cease being true if all sentient lifeforms were to perish strikes me as incredibly odd.

[Burrhus]

I have to say, the idea that mathematical truths would somehow cease being true if all sentient lifeforms were to perish strikes me as incredibly odd.

You have conflated propositional truth with existence.

[Baron_Corvo]

You have conflated propositional truth with existence.


Not at all. You can say what properties an odd perfect number would have, for example (equal to the sum of its divisors and not evenly divisible by 2) without having any way at all of knowing whether or not one exists.

[Burrhus]

Not at all. You can say what properties an odd perfect number would have, for example (equal to the sum of its divisors and not evenly divisible by 2) without having any way at all of knowing whether or not one exists.

Pollinosis: Mathematical Platonism is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices.


Originally Posted by Pollinosis
I have to say, the idea that mathematical truths would somehow cease being true if all sentient lifeforms were to perish strikes me as incredibly odd.

I was responding to Pollinosis' conflating existence with truth as demonstrated in the two quotes above.

[Pollinosis]

I was responding to Pollinosis' conflating existence with truth as demonstrated in the two quotes above.

I was merely pointing out the absurdity of reducing mathematical objects to thoughts in the minds of sentient beings.

[Tellurocrat]

I think there is a basic fact of quantity which lends reality to mathematics. Without sentient minds, mathematical propositions would have no actual existence. They would not cease to be true - saying they are true or false when they have no existence wouldn't make sense; quite simply their truth, without their existence, would be irrelevant and inapplicable.

The experience that I was actually exploring something with its own existence that had been sitting there waiting to be discovered for eternity before Benoit Mandelbrot found it (as opposed to telling my computer to render something that had been defined into some kind of contingent existence by him) was and is too hard to shake.
Similar experience is triggered during inspiration, perhaps during writing a story, a fictional event, a fictional metaphysics - you suddenly SEE how and why it works. It doesn't mean you're actually seeing anything real. It just feels that way.

[Burrhus]

I was merely pointing out the absurdity of reducing mathematical objects to thoughts in the minds of sentient beings.

Pollinosis: Mathematical Platonism is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices.

Originally Posted by Pollinosis
I have to say, the idea that mathematical truths would somehow cease being true if all sentient lifeforms were to perish strikes me as incredibly odd.

I can only respond to what you post. First you assert the independent existence of mathematical 'objects', then you assert their independent truth without clearly delineating any distinction between the two.

That's conflation.

Those of us who are simply humans and not beings, and cannot discover within us any mind stuff are loathe to confer existence on so-called objects even more ephemeral than the hypothesized mind.

[Macrobius]

Those of us who are simply humans and not beings, and cannot discover within us any mind stuff are loathe to confer existence on so-called objects even more ephemeral than the hypothesized mind.

So let's formalise that, editing out a bit of the emotional but irrelevant rhetorical shading and downgrading 'can discover' (ability) to material fact of extensionality of discoveries -- 'does not discover'. Let the relation E be xEy 'x is more ephemeral than y', the monadic predicate Mx 'x is Mind Stuff', xDy 'x discovers y within x itself', Hx 'x is simply Human', and xLy 'x loathes to confer existence on y'.

Your claim seems to amount to:

D(y,z) = 'A person y who discovers something z more ephemeral than mind stuff within that person'

D(y,z) = {y | yDz and for all w (if Mw then zEw) }

P(y) = 'A person who does not find mind stuff within that person'

P(y) = {y | for all z, yDz then not Mz }

Putting it together, your claim thus seems to be:

For all humans y, satisfying Hy, and any z in the universe of discourse,

if P(y) and D(y,z) then yLz. (*Burrhus)

Loosely: 'All materialists are loathe to credit entities less material than mental objects with existence'.

Seems unlikely. I await your deductive or inductive demonstration of your assertion. However, I think it unlikely you can find a rational ground for it of any sort, for any but a trivial semantics for 'loathes to confer existence on'. (By trivial I mean such as 'I tend to feel nauseous when I confer existence on things, and have been behaviourally trained not to' -- which is not scientific evidence of anything except your conditioning.)

Your claim seems to be convertible with 'I am a materialist and I deny extra-mental immaterial objects, but can't say why'. It is probably strictly false in its original less subjective phrasing, or rather meaningless since you don't have access to everyone's mental states and claiming you do contradicts materialism in the same breath.

Nominalists may construe my sets as virtual classes, in the spirit of Quine's New Foundations.

[Burrhus]

Loosely: 'All materialists are loathe to credit entities less material than mental objects with existence'.

Your claim seems to be convertible with 'I am a materialist and I deny extra-mental immaterial objects, but can't say why'. It is probably strictly false in its original less subjective phrasing, or rather meaningless since you don't have access to everyone's mental states and claiming you do contradicts materialism in the same breath. .

What mental objects?

What extra-mental immaterial objects?

Of course I can't say why if by why you are asking for evidence. But you can't say why you affirm the existence of mental or extra-mental objects by the same criterion.

We all reason from within some set of indemonstrable metaphysical assumptions. Mine are different from yours.

I suspect that the problem here is that you, incorrectly, assume that since I am a materialist that I hold no metaphysical beliefs, that is to say, beliefs about the nature of reality that cannot be demonstrated by recourse to evidence. But I do.

I believe that the universe is purely material, eternal and lawful. You clearly believe that the universe is both material and immaterial. As you are a theist, I will assume that you also believe that it is both temporal and subject to the undetermined will of some immaterial entity.

Discourse emanating from within disjunct paradigms is fruitless. Discourse about disjunct paradigms is not necessarily so.

[Macrobius]

What mental objects?


Whichever ones you were using as your standard for operational comparison, on your way to loathing mathematicals. What I pointed out is you can have no such operational way to deny them, since you chose a criterion that you simultaneously don't believe exists. This is like picking a kilogram of Iridium in Paris to measure mass, when you have verified there is no such kilogram.


What extra-mental immaterial objects?


You will need to explain the status of things you deny exist in your metaphysics for me to give a reasonable answer. Do contrafactual objects that do not exist in this world have a shadowy existence in possible worlds, or do they have an absolute nihilistic non-existence that we can't make meaningful predications about? In the latter case, why did you invoke them, and in the former why do you deny them?


Of course I can't say why if by why you are asking for evidence. But you can't say why you affirm the existence of mental or extra-mental objects by the same criterion.


Absolute relativism will not help much in argument, if you intend to persuade anyone else. However, I will grant that relativistism may seem irrefutable to a solipsist who shares no communication or metaphysical principles with anyone else. I'd also say that is an anomalous and rare condition, not the norm you seem to assert, even for 'humans who aren't beings'. Pretending unlikely things are ubiquitous is no more appealing to me than extra-mental mathematical objects are to you. I cannot think of any good reason why you should expect me to be persuaded by such chicanery. A priori, there is no reason to pick your 'my meaningless relativism the measure of everything' over my (or any reasonable) alternative.


We all reason from within some set of indemonstrable metaphysical assumptions. Mine are different from yours.


The possibility of knowledge does encode some irrefutable metaphysical notions in everyone, and there is remarkable agreement as to them between 'systems' or cultures. Modern man is unique among all human cultures for specifically blinding himself in certain ways and pretending everyone is equally blind. However, to extent he carries out this pretence convincingly, he renders himself incapable of the technological and scientific progress on which he bases his pride of claim.


I suspect that the problem here is that you, incorrectly, assume that since I am a materialist that I hold no metaphysical beliefs, that is to say, beliefs about the nature of reality that cannot be demonstrated by recourse to evidence. But I do.


I am not familiar with the metaphysics behaviourists traditionally espouse, however I take it to be some for of phenomenalism that gives primacy to sense data and constructed perception derived from sensation, together with the functions prescribed by genetics, as primary inputs. I take Materialism to be a denial that there is any objective formal content in things at all -- roughly the assertion that what they are composed of is a sufficient explanation of their being, without resort to essentialism, species, Forms, teleology, or the like. The status of efficient causes and intentionality is likely ambiguous and problematic in materialism.

I am sure we don't agree on metaphysical principles. As a physicist, I defend a moderate realist theory of knowledge -- the Aristotelian claim that extracting formal content from observed reality via the 'species individuum structure' (SIS) is a capability of Man, and underlies the possibility of an sort of Science -- in the sense of knowledge, theoretical and practical -- at all. Materialism and Sensism would both deny there is any formal content at all [essential form, species] to be extracted, much less that it is possible, with or without the aid of sense data. Between my view and the modern lies Cartesian Dualism, which bounds modern thought for all schools of philosophy and psychology since.

As to whether Plato is correct, I think we should first understand Aristotle and modern Science, then decide between Plato and a revised Aristotelianism. This procedure is less likely to lead down a blind alley -- namely one that discards Physics and Biology in the process.

Reasonable men share certain logical axioms, and while there is no such thing as a self-proving and self-evident metaphysical system with no assumptions at all, none the less the reasonableness of mathematics and science can be explained and justified to those who have not deliberately dehumanised (and post-rationalised) themselves in the way I described above.

However, this demonstration cannot be on the basis of the psyche alone, and specifically not on its contents when restricted to empirics and perceptions.


I believe that the universe is purely material, eternal and lawful. You clearly believe that the universe is both material and immaterial. As you are a theist, I will assume that you also believe that it is both temporal and subject to the undetermined will of some immaterial entity.


You may be on to something and certainly there is a way to express a disagreement with me. However, 'immaterial' will give us problems. As a physicist, I am aware of two meanings of the term 'mass' -- one is a 'quantity of matter' and thus measure of intertia, and the other is 'stuff' (corporeality, which includes non-penetration to the touch as a core intuition).

After Descartes, the two notions are combined in a metaphysical system but there is no particular reason, on Physics grounds, to insist that a mesoscale intuition about 'matter' being solid and resisting penetration -- a phenomenon of solid state physics now well understood -- should be combined with the medieval notion of a geometry of nature (matter as extensibility).

Matter (in the lowest, corporeal order) is nothing more and nothing less than the geometric substrate on which we hang perceptions. The question since Kant is whether it is created, or constructed, by a biological process in whatever you take the psyche to be. Only within the restricted confines of that metaphysical view (German Idealism) does it make sense to define materialism as we take it to be today. Roughly, it is a denial of 1/2 of Descartes, starting from a Kantian psychology. This is a very limited philosophical view, and does not embrace the full meaning of 'matter' even in Descartes, much less Aquinas or Newton.


Discourse emanating from within disjunct paradigms is fruitless. Discourse about disjunct paradigms is not necessarily so.

I'm not sure what a disjunct paradigm is but I think you mean dialogue between incommensurable belief systems. Macintyre has an excellent and historical discussion of this problem -- it has limitations but it is an effort at least -- in his Whose Justice? Which Rationality. It is not entirely hopeless, but I agree it is problematic. His solution is that if System S1 can explain the deficiencies of S2 by translating the problematics of S2 into the world view of S1, but the converse is not true, we have rational grounds for saying S1 is superior to S2. He claims the Aquinas-Aristotle tradition has done this relative to its existing competitors in the academic world. In a limited way, I agree with him on this, while leaving open an Augustinian or 16th/17th century Neo-Platonic philosophy critique, as well as deeper critiques based on more Traditional Science.

In short, radical relativism and insistence 'we can't argue' can be refuted by an end run: simply show that, for those willing to credit the claimed psychological and spiritual faculties, that one arrives at a superior understanding (in translation) of the other side's problematics. If you can solve the other side's problem *better* than the other side can, by staying in its own system, you can claim to 'embrace and extend' relativism in the same way Relativity embraces and extends the problems of Newtonian physics. Relativism becomes a limit of the larger, more comprehensive case, in which the participants deliberately ignore faculties and evidence that they in fact do have. No one is really convinced by this, barring a drop-dead argument such as superior progress in Science. When this fails to show on schedule, the faithful, mindless masses of Science-worshipers will evaporate faster than cannibals running for a missionary in a pot.

Science, to the extent it takes your view seriously, will render itself impotent. You are living on borrowed time. Buy some herbs.

[Ahknaton]

Similar experience is triggered during inspiration, perhaps during writing a story, a fictional event, a fictional metaphysics - you suddenly SEE how and why it works. It doesn't mean you're actually seeing anything real. It just feels that way.
I think that this experience of seeming to have an independent existence is stronger with mathematical truths than with, say a novel, because their internal consistency and so on is far more objective and the opportunities for their creators/discoverers to make arbitrary changes are a lot fewer, but I concede your point that this strong feeling isn't enough to prove that they really do have an independent existence. However a lot of the same arguments can be used to doubt the independent existence of material reality. Just because our sense perceptions seem to indicate an external self-consistent reality doesn't mean it actually exists, but for all intents and purposes we just accept that it does. Why should I show a greater degree of skepticism towards mathematical truths than I do towards external reality?

[Baron_Corvo]

I think that this experience of seeming to have an independent existence is stronger with mathematical truths than with, say a novel, because their internal consistency and so on is far more objective and the opportunities for their creators/discoverers to make arbitrary changes are a lot fewer, but I concede your point that this strong feeling isn't enough to prove that they really do have an independent existence. However a lot of the same arguments can be used to doubt the independent existence of material reality. Just because our sense perceptions seem to indicate an external self-consistent reality doesn't mean it actually exists, but for all intents and purposes we just accept that it does. Why should I show a greater degree of skepticism towards mathematical truths than I do towards external reality?

Good point. We know about physical reality because we perceive it through our bodies, but those are themselves part of physical reality. It's a sort of catch-22 situation (at least that's the best phrase I can think of this late at night).