The Platonic Mathematical World

[SerScot]

Zara,

Mathematics is true A Priori. Is there anything else that can make this claim?

[Avatar]

But all this is separate from the question of the objectivity of math. It does not get its objectivity from the physical world.

How can something be objective other than in the physical world?

1+1 objectively (physically) equals 2.

--A

[SerScot]

Avatar,

But that a posteriori rational. What makes math objective is that it is true a priori.

[Hashi Lebwohl]

Some might argue that that is not possible, SerScot, but I adhere to that belief, myself. The concepts of "1", "+", "=", and "2", which feed into the true statement "1 + 1 = 2" was true before we discovered it and it will continue to be true after we are all dead and gone. That which we call "1", "+", "=", and "2" are just our words or representations of those ideas. This is why I agree with Sagan's conjecture that mathematics will be how we enter into a dialogue with extraterrestrial species, should we ever encounter any. They may not know what we mean by "hello. who are you?" but they will recognize the Pythagorean Theorem when they see it.

[Holsety]

But all this is separate from the question of the objectivity of math. It does not get its objectivity from the physical world.

How can something be objective other than in the physical world?

1+1 objectively (physically) equals 2.

--A

But 1 chair + 1 chair = 2 chairs is a subjective accounting of the physical world. It's not like there's a predefined definition for what physical closeness, similarity in design, ownership or lack of, etc., makes a set of chairs "groupable." It's a matter of perspective, right? Why, even if we counted all the chairs we thought were in the world, we could've missed some, right?

I know this post probably comes across as stupid, but I decided not to say anything other than this sentence deprecating it in order to save time.

EDIT-On second thought I think this is just an articulation of what SerScott said, but I'm going to leave it here regardless as a just in case (?).

[Avatar]

What makes math objective is that it is true a priori.

A priori knowledge only holds up until experience contradicts it. The best we can say is probably that we believe it to be true.

Is math inductive? Or deductive? And if it is inductive, then can it still be an a priori truth? Can anything be true a priori?

--A

[Vraith]

Can anything be true a priori?

--A

Heh...well, it may be true a priori in some sense, "exist" in some way...but it isn't "knowledge," and isn't meaningful, really, until something capable of reason, experience, and empiricism exists.

[Avatar]

In other words no?

--A

[Vraith]

In other words no?

--A

No, not no.
I keep saying that it's different KINDS of real.
And that applies to that.
[and the other thing I keep saying is that my biggest objection
to much of the argument is the positioning [made popular by Plato] of the mathematical/ideal above the actual, particular, material instead of below it.]

[SerScot]

Avatar,

What makes math objective is that it is true a priori.

A priori knowledge only holds up until experience contradicts it. The best we can say is probably that we believe it to be true.

Is math inductive? Or deductive? And if it is inductive, then can it still be an a priori truth? Can anything be true a priori?

--A

How can Math not be a priori?

[Hashi Lebwohl]

I can see what SerScot is saying. Human beings did not "invent" or "prove" that 1 + 1 = 2 (and so on and so forth) but, rather, we "discovered" those mathematical truths even before we formalized any mathematical concepts. Now imagine some other intelligent species with a different physiology (specifically, maybe they don't have two arms and/or ten fingers) and psychological makeup from our own. They will arrive at exactly the same conclusion we did--1 + 1 = 2--even though they don't look like us or think like us. This means that that mathematical statement is true regardless of who discovers it and, as such, will remain true after we are gone. If this does not qualify as "a priori" then what does?

Arithmetic, the foundation of all higher mathematics, requires only hands-on investigation to verify, meaning that you don't have to rely on formal research (the Scientific Method) to reach your conclusions. Once you teach a three-year-old what basic counting numbers are, which is possible, you can teach them addition by telling them what "plus" means. "one block here, one block there, put them together--plus--and you get two blocks. one block plus two blocks is three blocks--one, two, three."

Everything higher than arithmetic can be considered "a posteriori" but even that is questionable. Go back to the aliens I mentioned earlier. Their terms for everything will be different but the Pythagorean Theorem will still be true for them. Calculus will probably be different because it is impossible to say how they find arc length or area under a curve....but it probably wouldn't be too different from our methods.

[Avatar]

How can Math not be a priori?

Is higher mathematics a result of deductive reasoning? Or inductive reasoning?

(Good description Hashi.)

--A

[Mighara Sovmadhi]

Is higher mathematics a result of deductive reasoning? Or inductive reasoning?

Mostly deduction, but also some more "constructive" but non-inductive/abductive (sometimes the word is "ampliative") reasoning. Good luck understanding the latter (we philosophers/mathematicians don't know what to make of it so much, yet, and some of us end up saying that it's "intellectual intuition"!).

[Avatar]

I feel that anything inductive can't be an a priori truth, because it isn't based on experience (ie knowledge) but on assumption.

--A

[Mighara Sovmadhi]

I suppose it's not like we count 2 + 2 = 4 a bunch of times and then extrapolate, "2 + 2 = 4, always, everywhere." Instead, the first time we do the calculation is a representation of what we will do whenever we as such calculate, so the output the first time has to be the same as the output later (on pain of the later output being that of a different, by definition, function--on pain of us talking about something else if we somehow got 2 + 2 = 16 or something).

But so the proof of an arithmetic statement isn't (unless you're doing a Russellian kind of logic) deduced from previous statements (like the definitions of the numbers computed) or induced from repeated examples of computation yielding a certain result. It's more like sight or hearing, except the information isn't impressed upon us from the outside, it comes from our innate mathematical power--it's proactive, not passive, knowledge.

[Zarathustra]

But all this is separate from the question of the objectivity of math. It does not get its objectivity from the physical world.

How can something be objective other than in the physical world?

--A

There are several ways. It can be the opposite of "subjective," which numbers and math clearly are, since they transcend individual minds who think them. It can also be objective in the sense that it's not biased, not the product of one's own wishful thinking or opinion, which math usually is, when we're not making mistakes. And it can be objective in the sense of "an object of thought," or "an object of consciousness."

Now this is where things start to get tricky, because an object of consciousness sure seems like something subjective. And it is. But since our entire reality is only ever encountered as an object of our consciousness, and we assume [in the natural attitude] or theorize [in philosophical introspection] that we achieve transcendence of ourselves in order to experience the world despite the subjective nature of our "objects of consciousness," then some aspect of those objects are conveying the truth of the world, the experience or appearance of reality.

And that's where numbers have their universal, objective, trans-personal, ideal existence ... or at least where we experience or encounter them in thought. We never really encounter them in the world, physically, as matters of fact.

It's pretty damn complicated, and I don't have the time or expertise to get into it--even if I could remember everything I've studied about it. But perhaps this weekend I'll take this opportunity to refresh myself on some phenomenology.

I will say this: our consciousness is "bifurcated" along a distinct division, so that it is always necessarily both consciousness-of and appearance-of. There is the part which is directed to an object, our intentionality, which makes our consciousness about something. And then there's that thing about which consciousness is concerned, that to which it is directed. That's the object of consciousness. These are phenomenological disinctions, made by Edmund Husserl. Noesis and Noema.

There are ideal, essential characteristics of each of these, which are necessary for any consciousness whatsoever, including consciousness of the world. And we can (through the phenomenological method) turn our consciousness-of (our intentionality) toward these essential characteristics. But in doing so, there is no reason to identify consciousness of an essence as the essence itself. It's not imaging, it's intuiting (like perception, except with ideal essences). Numbers are similar ... our consciousness of a number takes the form of a concept, but that concept is not itself the number. Numbers aren't created like concepts, but merely held in consciousness as a concept, while we're thinking of them. And numbers have necessary structures which we can hold in our consciousness, but likewise didn't create, nor are they identical to the consciousness of them. Their necessity, that is the necessity of those structures which relate numbers to each other (i.e. math), derives from the essence of the numbers, and not created by the mind that thinks them.

So, again, in that sense they are objective.

I feel that anything inductive can't be an a priori truth, because it isn't based on experience (ie knowledge) but on assumption.

"A priori" means "prior to experience." It's what can be said to be true without looking at the world at all. It's certainly not based on experience, and in fact is contrasted to that, which would be a posteriori, which can only be known to be true by looking at contingent facts.

[Avatar]

Exactly. How can we say definitively that something is true with knowledge of it?

If we haven't experienced something, our reasoning is an assumption. Our assumption may or may not be accurate, but we cannot know which until we experience it.

--A

[Mighara Sovmadhi]

But what is experience? The input of the senses? Any similar possible input, like if we could echolocate things? This is what the typical (and I include myself under this description) proponent of apriority will be denying: that all knowledge is based on external information being impressed upon us. We will say that some information is innate for us, or projected from within our innate cognitive structure. And this would be that which we know without experiencing it.

But you might define experience in terms of any consciousness at all.

[Avatar]

I don't think that any information is really innate in us.

There are certainly things that are innate, like a newborn's swimming reflex, but information? Knowledge? Where would it come from?

If you're talking about a collective unconscious, or genetic memory, I'm not sure if that could be classed as knowledge, even if it existed.

--A

[Vraith]

But what is experience? The input of the senses? Any similar possible input, like if we could echolocate things? This is what the typical (and I include myself under this description) proponent of apriority will be denying: that all knowledge is based on external information being impressed upon us. We will say that some information is innate for us, or projected from within our innate cognitive structure. And this would be that which we know without experiencing it.

But you might define experience in terms of any consciousness at all.

One might. [I'm understanding that sentence as: being/having consciousness, in itself, is an experience/experiential...even if one has no other senses.]
But one might also say that the ability to experience is a necessary prerequisite of consciousness.
And I think that's so.
Just for fun [fun for me, anyway...just gonna riff here for a bit]:
Assume that somehow a "consciousness" existed with no possible sensory/experiential apparatus at all, of any kind, and the conscious state had no necessary/existing connection to experience.
Imagine it somehow arrives at the apriori knowledge of "1," or some other...any other...mathematical fact/truth. From there, assuming it has incredible mathematical genius, it invents and proves the entirety of all mathematics. At that point: what knowledge does it actually have? What can it do with it? Does it know anything at all, CAN it know even one thing about the material/physical universe? Can it know even one thing that is NOT purely mathematical in nature? How?
That being strikes me as the ultimate example of [to be politically incorrect] "idiot-savant."
It seems relatively simple to me to move from physical to conceptual by abstraction and such. I don't see how to move in the other direction.

Now, I know in physics they now find material things because they are implied in the math. But that only happens because they have experience of other instances of math/material interactions.