l don't think there is any explanation as to how material objects, such as trees, can instantiate a mathematical number. — Sirius

On the one hand, as soon as some sufficiently capable intelligence wants to know of various attributions of “how many”, the material thing is that which instantiates both the object of such intellectual want in general and the means of satisfying it in particular.

On the other hand, material objects are not so much the instantiation of mathematical objects, but only serve as the occasion for an intelligence to instantiate by its own means, a method sufficient for that which it wants to know about those objects, from which arises the construction of mathematical objects.

What would be a reason for mathematical objects at all, if not for an intelligence that seeks judgement on certain kinds of relations, given between real, physical things? Perhaps, then it is neither the material object itself, not the intelligence itself, that instantiates mathematical objects, but it is merely the occurrence of natural relations between the two, that demands them.

It never was the question of explanation, but only the affirmative power of whatever it may be.
————-

It appears to me that our minds project mathematical concepts onto the world and shape our phenomenal experience for us. — Sirius

Yep, just like that. But the instantiation of them remains unexplained by the mere projection.

I don't think its a contradiction to say that there is an objective world and I just don't have access to it.

3. There are infinitely many statements that are necessarily true, independent of spacetime itself — Sirius

I dispute this. There may be infinitely many facts, but it does not follow that there are infinitely many true statements. Some facts just aren’t talked about.

At the very least you can say that there are infinitely many possible true statements, but I don’t think that requires an all-encompassing mind.

I dispute this. There may be infinitely many facts, but it does not follow that there are infinitely many true statements. Some facts just aren’t talked about. — Michael

If a fact isn't talked about (or held in some mind) is it a fact?

Also, there are infinitely many numbers, right? But there is a limit to the biggest number a finite brain can think of. No matter what notation is used, there will be a number that a finite brain just cannot conceive- it will be overwhelmed. The neurons/transistors/logic gates will be overwhelmed. So does that mean there aren't infinitely many numbers?

I dispute this. There may be infinitely many facts, but it does not follow that there are infinitely many true statements. Some facts just aren’t talked about.

At the very least you can say that there are infinitely many possible true statements, but I don’t think that requires an all-encompassing mind.

I take it you are willing to accept there can be infinitely many mathematical facts.

You take facts to be possible states of affairs, which must either be true or false. So given there can be infinitely many facts, an infinitely many of them are either true or false. By the law of non-contradiction, an infinitely many of them are true, likewise an infinitely many of them are false. An infinitely many true statements cannot be expressed by a universe which contains finite information. Hence, an infinitely many true statements exist independent of the universe

Note : Possible states of affairs can be false, for eg, 1+1=3 is F , given the usual notation. All actual statements are just true possible statements

If by possibly true you mean possible statements which are not actual but true, then it's a self contradiction. But if you mean the possible statement can be true or false, then you have said nothing. You are just repeating the law of non-contradiction, which is even true of actual statements.

You take facts to be possible states of affairs, which must either be true or false. — Sirius

Statements are true, states of affairs obtain. A statement is true if it describes a state of affairs that obtains, and false if it describes a state of affairs that doesn't obtain.

There are a finite number of statements but (possibly) an infinite number of states of affairs. Statements depend on "cognitive content" but (some) states of affairs don't.

Ideas clearly exist — Sirius

But do they? That's the question I'm raising. The point I'm making is that they exist in a different way than do phenomenal objects. As I said, the only way to communicate an idea is to explain it to someone capable of understanding it, and as you yourself say:

Cognitive content depends on the existence of a mind which can comprehend it — Sirius

But whose is the mind on which those facts depend? In what sense does that exist? And if it does exist, then why can't any empirical evidence be adduced to demonstrate it?

As you conclude, this 'mind' corresponds to 'an impersonal God' (although the 'impersonal' caveat might simply serve to defray the accusation of defending religious apologetics.) But fathoming that mind is another matter. That mind is not 'out there somewhere', it is not a phenomenal existent. So in what sense can it be said to be real? I think you're actually coming up against a pretty profound philosophical issue with deep roots in the tradition.

Forgive the length of the quotation below. It's from Dermot Moran's entry on Scotus Eriugena on SEP, slightly edited by me, and I provide it, because it's about the only source I'm aware of that makes this point in modal metaphysics. Note in some passages I've substituted 'to exist' for 'to be', to support the polemical point.

Eriugena proceeds to list “five ways of interpreting” the manner in which things may be said to exist or not to exist. According to the first mode, things accessible to the senses and the intellect are said to exist, whereas anything which, “through the excellence of its nature”, transcends our faculties are said not to exist. According to this classification, God, because of his transcendence is said not to exist. He is “nothingness through excellence” ¹

The second mode of being and non-being is seen in the “orders and differences of created natures” whereby, if one level of nature is said to exist, those orders above or below it, are said not to exist:

For an affirmation concerning the lower (order) is a negation concerning the higher, and so too a negation concerning the lower (order) is an affirmation concerning the higher.

According to this mode, the affirmation of man is the negation of angelic intelligence and vice versa. This mode illustrates Eriugena’s original way of dissolving the traditional Neoplatonic hierarchy of existence into a dialectic of affirmation and negation: to assert one level is to deny the others. In other words, a particular level may be affirmed to be real by those on a lower or on the same level, but the one above it is thought not to exist in the same way. If humans are thought to exist in a certain way, then angelic intelligences do not exist in that way.

The third mode contrasts the being of actual things with the “non-being” of potential or possible things still contained, in Eriugena’s memorable phrase, “in the most secret folds of nature”. This mode contrasts things which have come into effect with those things which are still contained in their causes. According to this mode, actual things, which are the effects of the causes, have being, whereas those things which are still virtual in the Primary Causes (e.g., the souls of those as yet unborn) are said not to be.

The fourth mode offers a roughly Platonic criterion for being: those things contemplated by the intellect alone may be considered to be, whereas things caught up in generation and corruption, viz. matter, place and time, do not truly exist. The assumption is that things graspable by intellect alone belong to a realm "above" the material, corporeal world and hence are timeless. — SEP, John Scotus Eriugena

My bolds. I think this reflects the hierarchical metaphysics of the 'chain of being' (scala naturae) which later fell into disrepute with the 'flattening' of ontology which characterises the advent of modernity. I think it stands for something real, but not empirically intelligible.

One of my arguments in OP is the well known indispensability argument offered by Quine-Putnam — Sirius

But what prompted that argument? Why were they obliged to defend the importance of mathematics in the first place? According to the IEP entry on same,

In his seminal 1973 paper, “Mathematical Truth,” Paul Benacerraf presented a problem facing all accounts of mathematical truth and knowledge. Standard readings of mathematical claims entail the existence of mathematical objects. But, our best epistemic theories seem to deny that knowledge of mathematical objects is possible.

So, what are our 'best epistemic theories', and why do they deny that such knowledge is possible?

Mathematical objects are in many ways unlike ordinary physical objects such as trees and cars. We learn about ordinary objects, at least in part, by using our senses. It is not obvious that we learn about mathematical objects this way. Indeed, it is difficult to see how we could use our senses to learn about mathematical objects. We do not see integers, or hold sets. Even geometric figures are not the kinds of things that we can sense.

This is just the point I'm making about sense in which number exists.

It goes on:

(Rationalist) philosophers claim that we have a special, non-sensory capacity for understanding mathematical truths, a rational insight arising from pure thought. But, the rationalist’s claims appear incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies.

My bolds. According to our 'best epistemic theories', mathematical insight should't be possible - because it contradicts naturalism! We are, after all, evolved animals, right? Nothing like that anywhere else in the animal kingdom! Hence Quine and Putnam's convoluted argumentation to defend something that should never have been called into question in the first place, but which seems to contradict the naturalist orthodoxy of the Academy. As SEP puts it,

Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects that aren’t part of the causal and spatiotemporal order studied by the physical sciences.[1] Mathematical platonism, if true, will also put great pressure on many naturalistic theories of knowledge. For there is little doubt that we possess mathematical knowledge. The truth of mathematical platonism would therefore establish that we have knowledge of abstract (and thus causally inefficacious) objects. This would be an important discovery, which many naturalistic theories of knowledge would struggle to accommodate.

(I would have more to say about the nature of the one mind you're proposing but it's already far too long a post so will come back to that.)

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1.This type of dialectic is encountered in apophatic theology, see for instance God does not Exist and What is the Ground of Being?

Also, there are infinitely many numbers, right? — RogueAI

Not in the Platonic sense. Numbers don't exist. Rather, when we say that there are infinitely many numbers we are just saying that we can (in principle) keep adding 1 forever.

Statements are true, states of affairs obtain. A statement is true if it describes a state of affairs that obtains, and false if it describes a state of affairs that doesn't obtain.

There are a finite number of statements but (possibly) an infinite number of states of affairs. Statements depend on "cognitive content" but (some) states of affairs don't.

I believe there is a fundamental disagreement between us regarding the ontological and logical status of possible states and actual states of affairs.

Here is what l hold onto


1. If A is a possible state of affairs, it was always possible. Possible states of affairs don't come into existence as a possibility.

2. Before this universe or any other universe existed, the possible states of affairs in reference to our universe and other universes out there existed, alongside all the possible states of affairs of universes which never came into existence.

3. Each possible state of affairs is either true or false. For mathematical statements, this is determined by logical necessity. As for empirical statements, they are assigned T/F in a brute manner, whether you believe in a God or not. Even if God assigned truth values to possible states of affairs related to the physical reality, it would be brute.

4. A true possible state of affairs is actual. What is actual, was never not actual. What is not actual is false. Otherwise, a possible states of affairs would be true and false.

Not in the Platonic sense. Numbers don't exist. Rather, when we say that there are infinitely many numbers we are just saying that we can (in principle) keep adding 1 forever.

What do 2, II, { { }, { { } } } , apple apple

all share ?

These objects don't belong in the list just because we wanted them to be in this list. The symbols are designed by us, but they represent something which isn't our construct, or else we would be able to do anything with them, such as equate 2 with 3, given the usual notation, bit this isn't possible.

If you are a strict nominalist who believes math is entirely our own construction, then you will need to explain why mathematical statements are not contingent and why do the laws of physics depend on them ?

As for infinity. Mathematicians never state, there are infinite possible numbers. Nope. They say

Take N = { 1, 2, 3... } , then N HAS an infinite number of elements

When we write lim x -- > inf 1 / x = 0 , we don't only mean as x approaches infinity, then 1 / x approaches 0, but when x approaches infinity, 1/ x does approach 0

The equality sign isn't there to represent an approximation. It isn't the job of philosophers to tell mathematicians what they can or cannot say.

I believe there is a fundamental disagreement between us regarding the ontological and logical status of possible states and actual states of affairs.

...

A true possible state of affairs is actual. — Sirius

Yes, there is a fundamental disagreement. My take is that truth is a "property" of expressions, i.e. speech, writing, thought, etc. As such, as you say, "true statements can only exist as cognitive content," given that speech, writing, and thought can only exist as cognitive content. But there's more to the world than speech, writing, and thought, and so that speech, writing, and thought can only exist as cognitive content isn't that the rest of the world can only exist as cognitive content.

But you seem to be arguing that truth is a property of other things (e.g. the states of affairs that speech, writing, and thought are about). It's not a position I agree with.

Even if I were to accept that truth is a property of these other things, note that these other things aren't statements. A true state of affairs isn't a true statement, and so that true statements can only exist as cognitive content isn't that true states of affairs can only exist as cognitive content. Your argument appears to equivocate.

I’m getting the feeling there’s been a bit of a leap-frog here.

If truth is mind-dependent even if about mind-independent objects, how are we assessing what is “true” in terms of definition?

If true statements are merely a mind communicating a mind-independent truth then none of this really works because we’re just discussing the subjectivity of lnguahe but if “true” is actually taken as a relation between the mind-independent world (ie any given object) and the mind who would potentially make a statement about it - ie that “true” just means the clearest transition from the world, to the mind, then the statements considered true require there to be nothing more than minds that can achieve a relation with the world to such a degree that other minds consider their statements to in fact carry that relation which is, to be very gentle, unhelpful in my estimation