I need your opinion on something. Let's say Platonism is true. That would mean math exists, because they're mental objects, in a mind, God's mind. Grant the materialist that minds are brains, physical. That would mean God too has a brain in which math exists i.e. God could be physical. Does God being physical/material affect theism in any significant sense? Speaking for myself, I'm totally ok with God being physical.

I know the discussion doesn't need an actual mathematician chirping in, but I will nevertheless.

Personally, I think math is invented by people to merely describe physical states of affairs of which some show exact correspondence with physical reality. I think Max Tegmark was on dope. — Prishon

Inventiveness in the subject has gone far beyond this; abstractions and generalizations move way beyond physical applications. But there is some truth in your statement. Babylonians measuring fields and Egyptians designing pyramids, etc. The jury is out on Tegmark. Some take him seriously. I don't.

As of today, I doubt that there is any maths left that has not been incorporated in some physics. — magritte

ArXiv.org receives hundreds of mathematics research papers each day. I would guess relatively few make it into physics.

“I believe that the only way to make sense of mathematics is to believe that there are objective mathematical facts, and that they are discovered by mathematicians,” says James Robert Brown, a philosopher of science recently retired from the University of Toronto. “Working mathematicians overwhelmingly are Platonists. They don't always call themselves Platonists, but if you ask them relevant questions, it’s always the Platonistic answer that they give you.” — Smithsonian Magazine, What is Math?

Creating relevant questions is an art form. Like a lawyer never asking a question they do not know the answer to. My observations are that working math people pay little attention to these issues. As for discoveries, when a mathematician conjures up definitions and relationships from wherever, perhaps as mere speculation or like a game, if there is a consistency to what is done then a slew of logical results may suddenly pop into existence, to be discovered by investigators. But there is that touch of creativity at the beginning.

know the discussion doesn't need an actual mathematician chirping in, but I will nevertheless. — jgill

If anyone CAN tell its a mathematician. Thanks for this great answer!

It's like that feeling you get when thinking it's friday and then realizing it's saturday.

If something is beyond space and time, then where could it be? — Corvus

That's nothing. Try this on for size.

1 X is within space & time. No!

2. X is beyond space & time. No!

3. X is within space & time AND X is beyond space & time. No!

4. Neither X is within space & time nor X is beyond space & time. No!

Where could X be?

If something is beyond space and time, then where could it be — Corvus

As usual almostly: :heart: (dont get me wrong though... ☺ ).

If something is beyond space and time, then where could it be? — Corvus

Only God knows, I guess.

Where could X be? — TheMadFool

Nowhere

I need your opinion on something. Let's say Platonism is true. That would mean math exists, because they're mental objects, in a mind, God's mind. Grant the materialist that minds are brains, physical. That would mean God too has a brain in which math exists i.e. God could be physical. Does God being physical/material affect theism in any significant sense? Speaking for myself, I'm totally ok with God being physical. — TheMadFool

Since Descartes, there is the apparent division between physical and mental - but I think there's something deeply the matter with that. Maybe it's a consequence of taking something that was originally a kind of explanatory metaphor - almost like an economic model, you might say - too literally. I question whether anything is 'purely' physical, or 'purely' mental - that view is a consequence of taking the model introduced by Descartes as if it is literally true, but it's an abstraction. A 'rogue metaphor'.

As regards the truth of 'mathematical platonism' - what interested me in the idea of mathematical platonism was the realisation that numbers don't come into or go out of existence (i.e. they're not temporally de-limited) and they're not composed of parts. But they're real, in that they're the same for all who can count, and they're among the elementary components of rational thought. When I hit on this idea, I thought I had seen why the ancients believed that numbers and geometric forms existed on a higher plane than ordinary things, which are always temporally delimited and composed of parts. At the time, that struck me as an epiphany, a significant Aha! moment. I don't know if I'm right in thinking that.

What struck me as interesting about the question is the sense in which numbers can be said to exist. Take any number - 7 will do. That is something that exists, you might say. But what you're looking at when you say that is a symbol. 7 can also be represented by VII, by 'seven', or in binary code, but what is being represented is always the same. The quantity that is represented by the symbol is a purely intelligible concept - it is only recognisable to an intelligence capable of counting.

The way I parsed that distinction, is to say that numbers (as universals) are real, but they don't exist.

There's some support for this distinction in Russell's discussion of the Problem of Universals, where he says:

We shall find it convenient only to speak of things 'existing' when they are in time, that is to say, when we can point to some time at which they exist (not excluding the possibility of their existing at all times). Thus thoughts and feelings, minds and physical objects exist. But universals do not exist in this sense; we shall say that they subsist or have being, where 'being' is opposed to 'existence' as being timeless. The world of universals, therefore, may also be described as the world of being. — Russell, The World of Universals

So - the sense in which numbers and universal are not physical, is that they reside in a realm of abstraction - but they're nevertheless real, in that they're not arbitrary or 'made up' (not that you can't invent imaginary numbers and systems, given the ability to recognise numbers.) It's important to realise the sense in which this relates to the meaning of 'transcendent' - not being 'spooky woo stuff', but truths that transcend time and space.

That is the 'formal realm' (i.e. the domain of numbers, shapes, and forms). But don't make the mistake of thinking it must be 'out there somewhere' - it does not literally exist. The urge to identify what is real with only what exists is one of the vices of modernity.

In the earlier matter-form (hylomorphic) dualism of the classical tradition, matter (hyle) receives form (morphe), and particulars are the combination of matter and form. So matter doesn't exist in its own right; in that view, the idea of something 'purely physical' is not intelligible. Maybe you could say that individual things are only real to the degree that they are an instance of a form. So it's not as if 'the physicall' is one thing, and 'form' another - they generally only exist together as a combination of form and matter. That's where it's very different to Cartesian dualism. (See this post on hylomorphic dualism in Aquinas.)

In that Smithsonian article I linked to on the nature of maths, it's said that empiricists will generally reject platonism. As the essay says, in today's culture, only what is physical is thought to exist. So obviously, that is incompatible with the platonist attitude. And this hails back to a titanic struggle in the history of ideas, in my view. That was how materialism became the dominant view. It has huge ramifications.

From the SEP article:

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 which aren’t part of the causal and spatiotemporal order studied by the physical sciences. 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. — SEP, Platonism in Mathematics

:up: Thanks a ton. I don't know if you'll recall a brief conversation we had a long while ago about whether the mind is a sensory organ or not. You were of the view that it is not but I insisted that the mind is a sensory organ just as the eyes, ears, etc. are, patterns being the mind's area of expertise. Since we seem to give what can be seen and hear the status of existence, if the mind were considered a sensory organ like eyes an ears, we would have to say that patterns, numbers being one of them, too exist. Interesting, no?

No. Numbers are not patterns. Bad idea. Reason and sensation are different faculties. Read that post I linked to on Aquinas, it says something extremely important and completely forgotten. Take some time to take that in, it's a deep idea. We're not engaged in banter here.

No. Numbers are not patterns. Bad idea. — Wayfarer

Not to contradict you but the number 1 is defined as the pattern (abstraction) in the following sets: {ghost}, {&}, {R}, {9}, {John}, you get the idea. What's common (the repeating pattern) is the one-ness.

Read that post I linked to on Aquinas, it says something extremely important and completely forgotten. — Wayfarer

:ok: Much obliged.

I'm objecting to the idea of reducing the faculty of reason to pattern recognition. I've seen people hawking that idea on philosophy forums and I think it is simplistic nonsense. For instance, the sequence of prime numbers is not a pattern. (There is a news story out there that some mathematicians have apparently found a kind of pattern in the sequence of primes, but it's disputed, and the fact that it's a story says something, because until now, it's always been understood to NOT be a pattern.)

Not to contradict you but the number 1 is defined as the pattern (abstraction) in the following sets: {ghost}, {&}, {R}, {9}, {John}, you get the idea. What's common (the repeating pattern) is the one-ness. — TheMadFool

It doesnt indeed contradict. Its simply wrong.

From the link you provided:

Sensible Form and Intelligible Form

“EVERYTHING in the cosmic universe is composed of matter and form.  Everything is concrete and individual. Hence the forms of cosmic entities must also be concrete and individual. Now, the process of knowledge is immediately concerned with the separation of form from matter, since a thing is known precisely because its form is received in the knower. But, whatever is received is in the recipient according to the mode of being that the recipient possesses. If, then, the senses are material powers, they receive the forms of objects in a material manner; and if the intellect is an immaterial power, it receives the forms of objects in an immaterial manner. This means that in the case of sense knowledge, the form is still encompassed with the concrete characters which make it particular; and that, in the case of intellectual knowledge, the form is disengaged from all such characters. To understand is to free form completely from matter.

“Moreover, 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.

“The separation of form from matter requires two stages if the idea is to be elaborated: first, the sensitive stage, wherein the external and internal senses operate upon the material object, accepting its form without matter, but not without the appendages of matter; second the intellectual stage, wherein agent intellect operates upon the phantasmal datum, divesting the form of every character that marks and indentifies it as a particular something.

“Abstraction, which is the proper task of active intellect, is essentially a liberating function in which the essence of the sensible object, potentially understandable as it lies beneath its accidents, is liberated from the elements that individualize it and is thus made actually understandable. The product of abstraction is a species of an intelligible order. Now possible intellect is supplied with an adequate stimulus to which it responds by producing a concept.”

So, the form (universals) individuates in objects (particulars). The senses, it seems, can't see past the particulars but the mind grasps the essences, another name for universals. :up:

I'm objecting to the idea of reducing the faculty of reason to pattern recognition. I've seen people hawking that idea on philosophy forums and I think it is simplistic nonsense. For instance, the sequence of prime numbers is not a pattern. (There is a news story out there that some mathematicians have apparently found a kind of pattern in the sequence of primes, but it's disputed, and the fact that it's a story says something, because until now, it's always been understood to NOT be a pattern.) — Wayfarer

:ok:

So, the form (universals) individuates in objects (particulars). The senses, it seems, can't see past the particulars but the mind grasps the essences, another name for universals. :up: — TheMadFool

No. The senses can look at ALL forms. There simply are no universals. The essence is no name for universals. The essence simply cant be defined.

The senses, it seems, can't see past the particulars but the mind grasps the essences, another name for universals. — TheMadFool

:ok:

The senses, it seems, can't see past the particulars but the mind grasps the essences, another name for universals. — TheMadFool

Dont put a thump up for a definition of the essence, please. You are playing God.

Where could X be? — TheMadFool

Intuition? or the pure reason?

You need to have some cognitive elements, visual or auditory, etc. in order to perceive space and time. Prior to this, there is no time and space. The Forms being unchanging, eternal, etc., cannot be anywhere else. — Apollodorus

I think I do. I am seeing clouds in the sky, and a hill below it. I am also seeing this text as I am typing.
I hear the sound of the cars passing outside on the road. But cannot find any forms. Well the only forms I normally see are in the junk mail for placing orders for clothing from the mail order companies.

No matter where I looked, the platonic forms were not found. Now I am guessing, they could be my intuition or pure reason.

Only God knows, I guess. — Prishon

Maybe one needs to do transcendental leap to be able to see them?

You need nous to see ‘em.

You need nous to see ‘em — Wayfarer

Would it be same as pure reason?

Not quite. The modern translation is ‘intellect’ but it’s a bit starchy to convey the gist. The Wiki entry is a good intro. ‘nous’ is preserved in vernacular English as being cluey or having a kind of insight (‘got nous, that bloke’)

Wiki on Nous is excellent actually.

Not quite. The modern translation is ‘intellect’ but it’s a bit starchy to convey the gist. The Wiki entry is a good intro. ‘nous’ is preserved in vernacular English as being cluey or having a kind of insight (‘got nous, that bloke’) — Wayfarer

It sounds nous is being suggested as equating part with pure reason in Wiki.

"As in Xenophon, Plato's Socrates frequently describes the soul in a political way, with ruling parts, and parts that are by nature meant to be ruled. Nous is associated with the rational (logistikon) part of the individual human soul, which by nature should rule. In his Republic, in the so-called "analogy of the divided line", it has a special function within this rational part. "

This is the passage that appeals to me:

n the Aristotelian scheme, nous is the basic understanding or awareness that allows human beings to think rationally. For Aristotle, this was distinct from the processing of sensory perception, including the use of imagination and memory, which other animals can do. This therefore connects discussion of nous to discussion of how the human mind sets definitions in a consistent and communicable way [this is where 'universals' come in to the picture] and whether people must be born with some innate potential to understand the same universal categories in the same logical ways [as Noam Chomsky argues].

Deriving from this it was also sometimes argued, especially in classical and medieval philosophy, that the individual nous must require help of a spiritual and divine type [which is the 'doctrine of illumination;]. . By this type of account, it came to be argued that the human understanding (nous) somehow stems from this cosmic nous, which is however not just a recipient of order, but a creator of it.

That basically is what I believe. Probably because of my Western memes. [Comments in brackets mine.]

By this type of account, it came to be argued that the human understanding (nous) somehow stems from this cosmic nous

What could be "this cosmic nous"?