The Platonic Mathematical World

[Orlion]

I've heard in his new book, Hawking claims that if something wasn't observed in the past, then that path is variable and not set. I need to get that book (and the one you mentioned, SerScot. Bohr and Einstein were quite the characters, They were usually more wrong then right, but Bohr's incorrect perceptions were a stepping block to Molecular Orbital Theory and Einstein helped popularize, at least briefly, the sciences, much like what Hawking does today).

[Vraith]

Was it Bohr and Einstein that said:
E: God doesn't play dice!
B: Stop telling God what to do!
Or was that E and someone else?
Anyway...yea, the "proof" of material things is in the pudding. If nothing was there to be perceived, [matter/energy/forces] we wouldn't [couldn't] have existed/evolved to perceive them.
non-material things/concepts...a lot more complicated [heh...or we wouldn't be arguing, I suppose].
I'm a little sad I didn't get even a chuckle on my M&M-Theory joke.
/weep

That's an interesting thing though, Orlion: The idea that we [if we had time travel] could alter the parts of the past that had no witnesses, but not the parts that did...[shades of Melenkurion cracking].

[Orlion]

Was it Bohr and Einstein that said:
E: God doesn't play dice!
B: Stop telling God what to do!
Or was that E and someone else?
Anyway...yea, the "proof" of material things is in the pudding. If nothing was there to be perceived, [matter/energy/forces] we wouldn't [couldn't] have existed/evolved to perceive them.
non-material things/concepts...a lot more complicated [heh...or we wouldn't be arguing, I suppose].
I'm a little sad I didn't get even a chuckle on my M&M-Theory joke.
/weep

That's an interesting thing though, Orlion: The idea that we [if we had time travel] could alter the parts of the past that had no witnesses, but not the parts that did...[shades of Melenkurion cracking].

There's no chuckle emoticon And if I recall, I'm pretty sure Einstein said that God doesn't play dice in response to Quantum theory... the uncertainty didn't sit well with him

[Hashi Lebwohl]

Without the human mind, there would be nothing to perceive mathematics. You can't say that it describes the universe, and then say that it exists independantly of the mind.

The thing that it describes may have an independant existence, (the volume of a sphere for example...spheres will have volume regardless of whether we are there to calculate that volume), but the description itself cannot exist without somebody to do the describing, or to describe it to.

--A

The description of something is not the thing itself. I refer you to the famous "La trahison des images" by Rene Magritte.

Mathematics exists without us but without us our descriptions and findings that we call "mathematics" would not exist. If I had to clarify, then I would say that numbers don't exist without us but geometric truths would still exist--the ratio of the circumference of a circle to its diameter is still constant, regardless of whether or not there are people willing to find that constan't value or give it a name.

Proof requires humans (or at least intelligent minds, even if they aren't human ones); truth does not.

[Avatar]

As I said...the phenomenon that it describes would exist. The description (the math) wouldn't.

Vraith, I think it was Hawking who commented on Einsteins claim that not only did God play dice with the universe, but he sometimes throws them where they can't be seen.

I've always liked Bohr. He once told somebody, "Your theory is crazy, but it's not crazy enough to be true."

--A

[Zarathustra]

Wow, once again there’s a lot to catch up with here. I keep forgetting to check in on this fascinating thread. (Feel free to PM me if I do this again!)

Time for thread necro. Zarathustra, interesting point. Are you saying Godel says we can understand the Universe but that we cannot prove our how we understand the Universe?

I don’t think Godel was addressing the universe. So the answer would be no, I’m not saying that. He was addressing formal systems. And even though we—beings in the universe—think about formal systems while we’re here, formals systems aren’t “in” the universe. They aren’t physical, tangible, or actual. (That’s not to say that they don’t have some reality … it’s evidence that there are nonphysical, immatieral, ideal features of reality which we can access).

Isn't the key to Godel's theorems (and the limit of their applicability) that they limit their assertions to the ability of formal systems to be proven with their own axioms.

His theorems dealt with both proof and completeness. I think both ideas are “the key.”

My mistake is equating proof with understanding?

I’d say that was a mistake, but I can’t speak for Godel would say. Certainly proof is a kind of understanding.

The more I read the more I don't believe the incompleteness theorems are about "Knowing" at all. They are about the ability to rigorously "prove" what we know. Not being about to prove something doesn't mean we can't know it. It also doesn't mean it does not have an existence independent of the human mind.

I’ll go along with that.

If something exists objectively, there must be proof for it. Subjective existence can be iin your head. Concrete, physical existence requires proof. It requires that everybody can see and experience whatever it is.

You’re using “proof” in a different way than SerScot (or Godel). There is mathematical/logical proofs (for a thereom in math or logic), and then there is empirical evidence (for a tangible object). Two completely different things. [This, I believe, I what SS meant here: "The key is whether it is provable a priori without obervation. That's the defintion of "proven" in mathematics. My wife explained why I was having difficulty with this over the weekend."]

However, I don’t have to construct a logical proof that “2” is a number, or that it's an ideal relation between objects. Anyone who understands the concept of plurality and enumeration knows intuitively (with apodictic certainty) that this ideal object is real, that it is objective in the sense that its meaning doesn’t depend upon my subjectivity, but rather transcends my subjectivity so that it’s the same exact ideal object when you’re thinking about it.

Godel, Escher, Bach is probably the most difficult book I have ever read. I recommend it to anyone and everyone.

I agree whole-heartedly.

If mathematics is a priori, it's synthetic a priori... peeps are still questioning how the synthetic "counts." Because if you "add" [1+2=3]...there are only 2 ways to begin proving it: Intuitive leap [so then ya gotta explain intuition] or by actually counting "something." [even if it's an imaginary or conceptual something] Which starts bleeding into a posteriori.

No. Math is tautological. You can’t have an equation without an “=” sign. This entails that whatever is on one side of that sign is derivable from the other side. That means it’s a priori. You don’t have to check any set of physical objects to determine that 1+2 = 3. You can do that with sets, if you want.

Intuition, in this sense, is the recognition of an a priori truth. (Kant talked about synthetic a priori to account for the effectiveness of science, I believe--not pure reason or math.)

I can agree about the geometry, but as you suggest, some branches of mathematics are based on our assumptions about the universe...different assumption, or different observations, will have different results.
Thing is, Serscot, that deductive reasoning relies on the accuracy of its postulates. It's possible to get a completely wrong answer using perfect logic, if the premises are mistaken.

You’re talking about the application of math to the real world. As long as we’re talking about a purely formal system, there is no issue of “accuracy of its postulates,” because there is no need to see if they match the real world. In this sense, the only time there's a "wrong answer" is when you apply the rules of the formal system incorrectly.

Starting point of this thread is that Mathematical Platoists, Godel among them, believe Mathematics has an existence and reality outside of the perceptions of the human mind. That this reality is why mathematics is so unreasonably effective

The objectivity of math (outside of our subjectivity) cannot be the explanation for the unreasonable effectiveness of math. This is still a great mystery! Possibly the only explantion for it is something like the Anthropic Principle combined with the Many Worlds interpretation of quantum mechanics. In other words: math seems unreasonably effective because intelligent life can only evolve in universes where such a relationship already exists; a universe that follows mathematical patterns has the right kind of symmetry for intelligence to evolve and contemplate such a rarity. But given enough time and enough universes, it’s bound to happen. It only looks unreasonable because we can’t see all the other universes which don’t follow mathematical patterns.

It's his position that the fundementals of Mathematics are discovered (Natural numbers and such) but the applications that arise from that discovery is invented. It means mathematics does have an existence outside of the human mind ...

Absolutely. I agree.

Mathematics does exist outside the perceptions and limitations of the human mind. Even if every human being in the world died right now, the statements in mathematics that are factually true (such as "for a given volume, a sphere has the smallest surface area") would continue to be true. Mathematics exists without us; we only uncover truths that were already there in the first place.

Yes.

And how, exactly, do we explain the apparent connection of the mathematical with the material?

That’s the most important question of all, I believe. The Anthropic Principle only accounts for why this universe seems to follow mathematical patterns. But even if pure random chance accounted for this fact, there seems to be a connection that runs deeper than correlation. And that connection is intelligence itself: the universe knowing that it follows these patterns. The question you pose is just as mysterious as the connection of the mental with the physical, the mind/body problem.

Without the human mind, there would be nothing to perceive mathematics. You can't say that it describes the universe, and then say that it exists independantly of the mind.

Correct. I think Vraith was pointing out that dichotomy. Math does more than describes the universe. It's more than a chance correlation. Once we're aware of it, we're connecting the "top and bottom" of reality. We're the most mysterious part of this whole thing.

If observation really creates the Universe, who was around to observe the big bang? Who was around for Eons after that as the Universe slowly became a place where it was possible for life to form? If observation really creates reality explain how we got here to what we observe today when most of it was created long before there was anything that could have observed it.

Prior to observation, the universe could have “existed” as a multitude or matrix of possibility, until consciousness formed the feedback loop that collapsed the quantum proxy waves into definite actualities. Kind of like seeding a crystal.

I've heard in his new book, Hawking claims that if something wasn't observed in the past, then that path is variable and not set.

Yes, that's kind of what I was talking about in the last paragraph above. I've got to get that book!

[Avatar]

However, I don’t have to construct a logical proof that “2” is a number, or that it's an ideal relation between objects. Anyone who understands the concept of plurality and enumeration knows intuitively (with apodictic certainty) that this ideal object is real, that it is objective in the sense that its meaning doesn’t depend upon my subjectivity, but rather transcends my subjectivity so that it’s the same exact ideal object when you’re thinking about it.

Not sure if I'm misunderstanding you here...the number "2" isn't a real object in and of itself. It only has meaning in as far as it describes a number of real items. Unless there are two objects, there is no "two."

--A

[Zarathustra]

However, I don’t have to construct a logical proof that “2” is a number, or that it's an ideal relation between objects. Anyone who understands the concept of plurality and enumeration knows intuitively (with apodictic certainty) that this ideal object is real, that it is objective in the sense that its meaning doesn’t depend upon my subjectivity, but rather transcends my subjectivity so that it’s the same exact ideal object when you’re thinking about it.

Not sure if I'm misunderstanding you here...the number "2" isn't a real object in and of itself. It only has meaning in as far as it describes a number of real items. Unless there are two objects, there is no "two."

--A

No, you're understanding me. I'm saying 2 is an objective--though ideal--object. It has an existence independent of my consciousness of it. And it has an existence independent of the individual examples we can point two where physical objects "partake in" or exhibit this relation. Its meaning does not depend upon the actual existence of two objects. Even if there was only one object in the universe--me--I could still contemplate another. How is that possible unless two has meaning beyond the existence of actual objects? How could we count ideas, for instance, if the meaning of "2" depended upon real objects? Are ideas real? If they're real, then why isn't "2" real? We can easily conceive of more numbers than the total number of protons in the universe. How is that possible if numbers are dependent upon objects for their meanings?

It comes down to whether you think relations are real or not. Not just the objects in the universe, but also how these objects stand in certain relations with each other. When you're done accounting for an object, when you're done describing it, it boils down to nothing more than a cluster of relations. The relations are the only real thing about it. Objects aren't things. They are clusters of processes. Matter itself is mostly empty space and a contortion of forces. These forces are nothing more than fields ... geometrically defined areas of space where certain predictable effects can happen. Substance itself is an illusion brought about by these forces, these fields. It would all be fluctuating chaos if not for the relations which pattern it all. If there is anything real to matter, it is adherence to mathematical patterns. You've got it backwards: objects wouldn't have meaning without numbers. In universes that don't follow mathematical patterns, there would be no objects.

[Vraith]

First, congrats on using the word apodictic, it's one of those words that just feels fun in my brain and mouth.

I agree with much of what you said about the relational...in fact, almost my entire interest in this thread is the mystery of how it works...and whether there are multiple kinds of different mysteries, or, as someone said "a calculus text is 2 pages of rules, and 2000 pages of examples." [roughly].

By which I mean: certain things that are require not just a different calculation, but entirely different kinds of math to deal with than other things.
And certain things that are calculable/real can't be material...and some are materially precise and definite but incalculable/indefinite mathematically.
So are the relations/indefinites/contradictions between maths/maths and material/material, and math/material the same mystery, with 2000 pages of example? Or are they completely different?
Heh...is the mind/body problem you mentioned an example of this mystery? OR: [this idea just popped in and made me laugh to myself] Is the mind/body, while remaining mysterious itself, the ANSWER to the other mysteries? [and is that what you meant by saying intelligence itself is the connection?] That amuses me: Just keep bashing maths and material together until you get a stable system and voila: intelligence will arise to look around and ask "is that a horse, or is someone banging coconuts together?"

[Hashi Lebwohl]

By which I mean: certain things that are require not just a different calculation, but entirely different kinds of math to deal with than other things.
And certain things that are calculable/real can't be material...and some are materially precise and definite but incalculable/indefinite mathematically.
So are the relations/indefinites/contradictions between maths/maths and material/material, and math/material the same mystery, with 2000 pages of example? Or are they completely different?

Look up the "Gabriel's Horn" problem. The volume contained inside the object is π but the surface area is infinite. Clearly, this cannot be real--no object can have infinite surface area in reality.

Now consider π itself. We don't know the exact value of this number and, in fact, we can never know it. All we can do is get closer approximations to its actual value.

Mathematics...reality...it's all a mystery.

[Fist and Faith]

Are ideas real? I can think of 2 ideas. Possibly neither the ideas nor 2 are real?

[Hashi Lebwohl]

Ideas are real--they are electrochemical patterns of neural activity in your brain; however, their reality will not be actualized externally unless you do something about them.

[Fist and Faith]

I think there's a difference between ideas and the electrochemical/neural medium in which they exist. Any reason a form of life based on other elements and processes than ours can't think and have ideas?

[Hashi Lebwohl]

Of course they can, but I am unfamiliar with their particular forms of biochemistry, leaving me limited with a human frame of reference.

[Fist and Faith]

My point is that the idea is not the same as the mechanism that thinks it. The idea of a pink, flying, six-legged cow is no more the electrochemical patterns of neural activity in my brain than it is the italicized words.

[Avatar]

I'm saying 2 is an objective--though ideal--object. It has an existence independent of my consciousness of it. And it has an existence independent of the individual examples we can point two where physical objects "partake in" or exhibit this relation. Its meaning does not depend upon the actual existence of two objects.

But you can't show me "two" unless there are two objects. The number only has meaning insofar as it refers to something. Obviously we don't rely on a physical object every time, but we tacitly accept that every time we say 2+2, it refers to 2 of something, even if the something is anything.

The concept of 2 is not an object, but it refers to an object. Even if the object is an atom or a unit of measurement. (Hmm...hang on...a unit of measurement isn't an object...it's another concept...so the objects can be concepts themselves...)

--A

[Fist and Faith]

(Hmm...hang on...a unit of measurement isn't an object...it's another concept...so the objects can be concepts themselves...)

I'm sayin'!

[Hashi Lebwohl]

My point is that the idea is not the same as the mechanism that thinks it. The idea of a pink, flying, six-legged cow is no more the electrochemical patterns of neural activity in my brain than it is the italicized words.

Good point. Kosh would say "the reflection never is itself" and so it is with an idea.

[Zarathustra]

I think there's a difference between ideas and the electrochemical/neural medium in which they exist.

Absolutely. Mental states have mental properties (subjectivity, intentionality, qualia, etc.) which the chemical processes themselves do not have. What you're objecting to with your statement is Identity Theory:

The identity theory of mind holds that states and processes of the mind are identical to states and processes of the brain. ...The identity theory of mind is to the effect that these experiences just are brain processes, not merely correlated with brain processes.

Some philosophers hold that though experiences are brain processes they nevertheless have fundamentally non-physical, psychical, properties, sometimes called ‘qualia’. Here I shall take the identity theory as denying the existence of such irreducible non-physical properties.

plato.stanford.edu/entries/mind-identity/

I fall into the camp of those "some philosophers" mentioned above who hold that experiences have non-physical properties which are irreducable to brain states--though I'm not a dualist! I'm a neutral monist.

That amuses me: Just keep bashing maths and material together until you get a stable system and voila: intelligence will arise to look around and ask "is that a horse, or is someone banging coconuts together?"

It amuses me, too, especially the way you put it.

[Fist and Faith]

I think there's a difference between ideas and the electrochemical/neural medium in which they exist.

Absolutely. Mental states have mental properties (subjectivity, intentionality, qualia, etc.) which the chemical processes themselves do not have. What you're objecting to with your statement is Identity Theory:

The identity theory of mind holds that states and processes of the mind are identical to states and processes of the brain. ...The identity theory of mind is to the effect that these experiences just are brain processes, not merely correlated with brain processes.

Some philosophers hold that though experiences are brain processes they nevertheless have fundamentally non-physical, psychical, properties, sometimes called ‘qualia’. Here I shall take the identity theory as denying the existence of such irreducible non-physical properties.

plato.stanford.edu/entries/mind-identity/

I fall into the camp of those "some philosophers" mentioned above who hold that experiences have non-physical properties which are irreducable to brain states--though I'm not a dualist! I'm a neutral monist.

Yeah, I agree entirely.

But I'm talking about yet another step. I'm saying there's a difference between the process of the mind and the idea it's thinking.