Mathematics and abstract reality

[becomingagodo]

Mathematics is not a language, although it uses a language. Some mathematician where platonist, Cantor, Godel, Ramanujan are good examples. And yet we have people who say mathematics is made up or just a language.

Is their a abstract reality?
If maths was a language or a invention of the human mind, then when does a mathematical statement become true. When the human mind invents it?, and if it is a invention then can the person invent something that is false when it is true? Even then some mathematics cannot be done in the mind, infinty and mandelbrot set show mathematic can't be all done in your mind or created in the mind.

Penrose argues that the mandelbrot set is true always and because it involves infinte then it can't be created in the human mind i.e. math is separate.

I blame applied mathematician and philosophers for the view that mathematics is just a language.

Mathematics shows something deep about reality. A good example is quantum mechanics, which is being used to prove RH. Uncertainty in mathematics is being linked to uncertainty in quantum physics. I use these two example as they show physics helping mathematics, however at first glance prime numbers would seem to have nothing to do with quantum mechanics.

Also, Niels Bohr argued that the atom is a physical object so removed from reality that anything in reality can't explain it. That you should abandon any picture of reality, and that reality itself can only be shown through mathematics. You can go further and argue that reality is mathematics. In the newscientist magazine a article was arguing that mathematics is reality.

Pure mathematics since Riemann has became more then equations, one mathematician said "When a flower opens, it makes no noise". At the end of the day abstract reality will show something about reality itself. Instead of showing you the truth, it will tell you the truth. However, maybe a select few can only hear the voice of mathematics. Since, both Ramanujan and Cantor did say they heard voices showing them mathematics.

[nescafe]

Max Tegmark (http://space.mit.edu/home/tegmark/toe.html) should be right up your alley.

[Gregory]

Mathematics is not a language, although it uses a language. Some mathematician where platonist, Cantor, Godel, Ramanujan are good examples.

And... is the Continuum Hypothesis Platonically true or false?

For that matter ... why should I care if Cantor, Godel, and Ramanujan are Platonists? Do you think their being Platonists makes it true?

Also, what does it mean for a set to be "true?"

[blobru]

Max Tegmark (http://space.mit.edu/home/tegmark/toe.html) should be right up your alley.


Hey, just what I was looking for, thanks! This topic came up in this thread: The Simulation Argument (http://forums.randi.org/showthread.php?t=95100&page=3); I wanted to explore whether "reality" is "analog" and simulation digital, if math is a priori or a posteriori, but couldn't find anything like online; Tegmark's External Reality (ERH) and Mathematical Universe (MUH) Hypotheses look like a great place to start... :)

[Dancing David]

Um, mathematics is a self referencing set of symbols. It has no meaning outside of reference to some sort of referent.

that is why it is like language.

We can approximate reality and decide which approximation has the best fit to the observation of reality. That does not mean the approximation is reality.

So where was tensor calculus before it was 'discovered". Did it exist is some Waiting Rooom of the Abstracted?

[FireGarden]

So where was tensor calculus before it was 'discovered".

Compressed inside the rules of maths?

Tensor Calculus isn't necessarily available in every possible maths. Just like the concept of pi (ratio of circumference/diameter) isn't available in every geometry.

I can set math problems without first knowing the answer. The answer is strongly implied in the setting of the problem. In that sense, it is not absent... merely undiscovered.

Where is it until discovered? Where can it be except in the problem?

The same is the case with a new mathematical theorem. It was always implied by the rules of maths.

[drkitten]

Mathematics is not a language, although it uses a language. Some mathematician where platonist, Cantor, Godel, Ramanujan are good examples. And yet we have people who say mathematics is made up or just a language.

Is their a abstract reality?

Not known. (As in, "how the hell did you expect to get an answer to this question, you dolt?")

In general, the constructivists seem to have won. Godel -- for all you admire him -- seems to have been influential in this. The fact that accepted axioms of mathematics demonstrably underconstrain the world (for example, the Axiom of Choice is independent of ZF, and the GCH is independent of ZFC) suggests that we need additional axioms. But since there appears to be no way to empirically vailidate either the GCH or its negation, there is no way to decide if the "abstract reality" follows the GCH or not.

Similarly, the fact that you can have consistent mathematical models with incompatible axioms (such as the various flavors of Euclidean and non-Euclidean geometry) suggests that neither Euclidean nor non-Euclidean geometry is "true" as a description of reality.

What's left, then, is the constructivst view. Axioms are chosen for convenience and descriptive power, theorems follow as consequences of axioms, and mathematics is the process of exploring those consequences.

[becomingagodo]

In general, the constructivists seem to have won.
Popularity is a sign of nothing. Mathematician are really confromist and don't like the concept of infinty. Even then some mathematician can't see the platonic world, only a few can.
The fact that accepted axioms of mathematics demonstrably underconstrain the world (for example, the Axiom of Choice is independent of ZF, and the GCH is independent of ZFC) suggests that we need additional axioms.
The ZF set theory is rubbish. Espically Naive set theory. The Axiom of choice has been critized and some don't accept it. We don't need additional axioms, we need a new theory.

Similarly, the fact that you can have consistent mathematical models with incompatible axioms (such as the various flavors of Euclidean and non-Euclidean geometry) suggests that neither Euclidean nor non-Euclidean geometry is "true" as a description of reality.
Axioms are stupid, however I guess you would need more abstract axioms. Quantum mechanics is incompatible with Einstein theory of general relativity.

What's left, then, is the constructivst view. Axioms are chosen for convenience and descriptive power, theorems follow as consequences of axioms, and mathematics is the process of exploring those consequences.
First off, don't give up like a coward at the first sign of danger, would be another way. Secondly, it would be good to explore what is causing Godels problem, you know it might be logical or part of the universe of logic itself. I don't really see a problem, well unless your a robot who wants to do mechanical proofs. Thirdly, now quantum mechanics is strange and has problems, this obviously connects to the maths of Godel.

I know I am biased, however mathematician are mostly idiots. It kind of like Gauss the person who is considered the best mathematician ever(I don't agree), he only collaborated with great mathematician. Now part of the reason is proberly because most mathematician are not that smart. It's like complex numbers, most mathematician didn't like it calling it imaginary, when Gauss started using complex numbers the mathematician slowly followed behind.

[Jekyll]

The ZF set theory is rubbish. Espically Naive set theory.

Zf axioms =/= naive set theory.

Axioms are stupid,
Thirdly, now quantum mechanics is strange and has problems, this obviously connects to the maths of Godel.
Perhaps some understand of formal reasoning would stop you from assuming that all things you don't understand are connected.

[drkitten]

The ZF set theory is rubbish. Espically Naive set theory.

Proving that you don't understand either, I'm afraid. ZFC is more or less the de-facto formalization of all mathematics today, and it replaced naive set theory because naive set theory was demonstrably paradox-ridden.

The Axiom of choice has been critized and some don't accept it.

Almost no one working in mathematics today would agree with you. We had that discussinon fifty years ago and moved on.


We don't need additional axioms, we need a new theory.

And this gets the big "yeah, sure, whatever."



First off, don't give up like a coward at the first sign of danger, would be another way.

Oh, yeah, because formal set theory is so dangerous. Did you know that there are thousands of mathematicians every year who come down with White Lung Disease from chalk dust exposure?

Secondly, it would be good to explore what is causing Godels problem, you know it might be logical or part of the universe of logic itself.

We've already explored it, quite thoroughly. It's a consequence of logic. Been there, done that, bought the T-shirt.

Thirdly, now quantum mechanics is strange and has problems, this obviously connects to the maths of Godel.

Not in the slightest, as far as we know.


I know I am biased,

No, you're simply ignorant. You don't know enough to reach the lofty heights of "biased."

[becomingagodo]

Proving that you don't understand either, I'm afraid. ZFC is more or less the de-facto formalization of all mathematics today, and it replaced naive set theory because naive set theory was demonstrably paradox-ridden.
The point I was trying to make is that Naive set theory is bad, ZFC is better but still bad. However, their is a better set theory. Also, formalization or rigour as mathematics call it is a sign of nothing, I remeber when they had to change the concepts of infintisimlar so it is described as limits. After that I don't trust any mathematician, or popular views.

Almost no one working in mathematics today would agree with you. We had that discussinon fifty years ago and moved on.
Popularity is a sign of nothing. Okay to put it another way, Axiom of choice is independent of ZFC, however ZFC is rubbish.

Oh, yeah, because formal set theory is so dangerous.
A history lesson on the critizism of Cantor would suggest so. Mathematicians heavily critized Cantor, calling his mathematics a disease and a corruptor of youth.

We've already explored it, quite thoroughly.
I'm mean where does uncertainty come from. Gregory Chaitin for example.

Not in the slightest, as far as we know.
Again, Gregory Chaitin suggested this and Hawkings.

No, you're simply ignorant. You don't know enough to reach the lofty heights of "biased."
I know history and I know philosophy. I don't see my views changing on maths, even when I have got a degree in higher mathematics.

[drkitten]

I remeber when they had to change the concepts of infintisimlar so it is described as limits

No, you don't. You're not that old. And you're misremembering what you've been taught.

I know history and I know philosophy.

Not as far as I can tell. You make too many elementary mistakes, you can't evaluate documents objectively, and you confuse too many concepts to read your own biases in.

I don't see my views changing on maths, even when I have got a degree in higher mathematics.

That's all right. I don't think I see you getting a degree in higher mathematics.

[Gregory]

So "ZFC is rubbish" ... uh huh. And which axiom system would you suggest?

[FireGarden]

So "ZFC is rubbish" ... uh huh. And which axiom system would you suggest?

I'm more impressed with Beco when he asks honest questions. Like when he asked for help with induction:
http://forums.randi.org/showthread.php?t=95514

Attitudes like that show intelligence.


Pretending you can go (within a month) from needing help with induction to being able to comment with authority on ZFC is not a sign of intelligence.

[Complexity]

becomingagodo recently asked for help on simplifying fractions, claimed that he could learn calculus in a month, gets most of his information about mathematics from collections of quotes by mathematicians, doesn't have the background to make any sense out of popular books about math, wants to be the greatest mathematician in history, and loves to pass judgement on things and people that he has heard of but is less than clueless about.

He is young (around 18), behind where he should be in his mathematics education, and believes that mental illness and suffering are the key to genius.

I have gone back on forth on whether he is salvageable. I think he might be if he'd stop pissing people off, drop the attitude, get the help he needs, and start being open to learning things, but those seem to be things that he's unwilling to do, at least at this point.

I'll keep watching and posting occasionally, but I'm not going to invest much time or concern in him at this point. Been there, done that. I'll change my mind if his behavior warrants it.

[drkitten]

I have gone back on forth on whether he is salvageable. I think he might be if he'd stop pissing people off, drop the attitude, get the help he needs, and start being open to learning things, but those seem to be things that he's unwilling to do, at least at this point.

Those are also exactly the traits that will keep him from completing his degree. He's unwilling to do the necessary basic reading, and his understanding of the material he has read is of negative value. He will botch the first problem set he is assigned in any course requiring abstract reasoning and conceptual understanding, and rather than try to understand the corrections his instructor gives him, he will simply announce that the theoretical basis behind the concepts is "rubbish."

This will last about two rounds before his instructor writes him off as being uneducatable and he is quietly dropped from the program.

[Complexity]

Those are also exactly the traits that will keep him from completing his degree. He's unwilling to do the necessary basic reading, and his understanding of the material he has read is of negative value. He will botch the first problem set he is assigned in any course requiring abstract reasoning and conceptual understanding, and rather than try to understand the corrections his instructor gives him, he will simply announce that the theoretical basis behind the concepts is "rubbish."

This will last about two rounds before his instructor writes him off as being uneducatable and he is quietly dropped from the program.


That is exactly where he's heading right now. He can change, but I don't think he will. It is like watching an imminent train wreck in slow motion.

[FireGarden]

Anyway....
To get back on (or close to) topic.

Where was calculus before it was discovered?
How much infomation is in a set of axioms? Do you have to count up all the theorems that can be deduced?

[Dancing David]

Anyway....
To get back on (or close to) topic.

Where was calculus before it was discovered?
How much infomation is in a set of axioms? Do you have to count up all the theorems that can be deduced?

Now just so you know I do understand that the concept has an intuitive feel. We find that the Rocky mountains existed before we became aware of them and things like that. In practicing aikido or 'making' contra dances there is a sense that "I did not invent this, I just described what was already there". So even though you may find a move that is new to you or create a set of moves that was never made before there is that feeling. It was already there before I went and deliniated it.

But this is the point that i am trying to make and it applies to language, math, aikido and contra dancing equally.

"It is inherent as an expression of the rules", also that there are a certain numbers of general motions of the arms and legs and two bodies to create the movements under the principles applied to aikido, given sets of four people there are certain combinations of dance moves that make flow to a particular person.

Before there can be the rules , before there can be the aiki movements before there can be the figures in the set: there are people.

The rules are generated by people, the movements are dictated by the human body, the dance moves are social constructs. So even though it may feel 'this was there any one could have found it', where is it if there is no one?

It then becomes one of a set of potential maths in expression, a potential set of movements of a potential body.

I am saying that it is deopendany upon the contingent history. It may appear to exist in the absence of humans and our brain structure, but that is like talking about the rules of physics in the absence of a universe. It can be done but it is rather idiomatic. The only reason we can talk about it is because there is an external referent to reference.

IMNSHO

[drkitten]

Where was calculus before it was discovered?

Sitting next to Hamlet in God's Library of Unwritten Books. He doesn't use the Library of Congress filing system, you see.


How much infomation is in a set of axioms? Do you have to count up all the theorems that can be deduced?

That's actually a very deep question. Obviously you can't count the theorems, since any interesting axiom set will yield the same (infinite) number of theorems. I don't think there's any widely-accepted answer to your question.

[Yiab]

Where was calculus before it was discovered?

"Calculus" didn't exist before it was "discovered" (in so much as abstract concepts in an intersubjective arena ever "exist"). The things we use calculus to describe still worked the same way, though.

How much infomation is in a set of axioms?

That depends on the language in which the axioms are stated, how many axioms there are, how redundant the axioms are and what their specific content is. Also, it depends on whether or not we are dealing with first- or second-order logic (or whatever higher order you like).

Do you have to count up all the theorems that can be deduced?

No. In any interesting case, the number of theorems which can be proven is equal to $|\mathcal{L}|\aleph_0$ where $\mathcal{L}$ is the language in use and $\aleph_0$ is the smallest infinite cardinal (for the right definition of "interesting", of course).

[FireGarden]

In practicing aikido or 'making' contra dances there is a sense that "I did not invent this, I just described what was already there". So even though you may find a move that is new to you or create a set of moves that was never made before there is that feeling. It was already there before I went and deliniated it.

I'm more than willing to accept that this applies to things other than maths.

So even though it may feel 'this was there any one could have found it', where is it if there is no one?

Why should a reality have to exist without depending on anything else?

It may appear to exist in the absence of humans and our brain structure, but that is like talking about the rules of physics in the absence of a universe.

But then what is an abstract reality?
Isn't it just a world in our thoughts? And within that world, things are discovered which are already defined -- "inherent in the rules".

I could agree that the abstract reality we call maths was itself created when maths was invented. But it exists now. And I can't escape the notion that once the axioms are stated, they define a world of thought. All that is left is to explore it -- or build a new one from a different set of axioms.

[Dancing David]

I'm more than willing to accept that this applies to things other than maths.



Why should a reality have to exist without depending on anything else?

That is assuming math is a reality.

Reality is reality, those things that appear to exist in and of themselves. If there was no one to study the relations then where is math?



But then what is an abstract reality?
Assumed?

Isn't it just a world in our thoughts? And within that world, things are discovered which are already defined -- "inherent in the rules".

In your thoughts.
Biochemical interactions.


I could agree that the abstract reality we call maths was itself created when maths was invented. But it exists now. And I can't escape the notion that once the axioms are stated, they define a world of thought.

Potential thought. there are also many 'wrong' ones along the way. So where are they in that 'reality'.

All that is left is to explore it -- or build a new one from a different set of axioms.

Where are they outside of the biochemical brain?