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Author | Topic: Mathematics and Nature | |||||||||||||||||||||||||||||||
RAZD Member (Idle past 1436 days) Posts: 20714 From: the other end of the sidewalk Joined: |
Just out of interest, why? Always is a very strong word. Because math is a purely - and idealized - intellectual construction that does not exist outside the mind. Take the counting of objects as an example: no two objects are really completely, purely 100% identical, so there is only 1 of anything. We "fuzzy" the definition of "object" to include a set of extremely similar objects - idealize them - to apply the intellectual concept. We could argue about the concept of a plane as a surface, but let's cut to the quick and take {the still simple yet one level more complex concept of} the mobius strip: no such thing exists in reality. You know my answers to the other questions - they have been extensively covered elsewhere, so we do not need to take this ot - lets keep this one about just the maths eh? we are limited in our ability to understand by our ability to understand RebelAAmerican.Zen[Deist
... to learn ... to think ... to live ... to laugh ... to share.
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Brad McFall Member (Idle past 5063 days) Posts: 3428 From: Ithaca,NY, USA Joined: |
You sound like "Frege" to me. Sorry I have no more time tonite!!
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cavediver Member (Idle past 3674 days) Posts: 4129 From: UK Joined: |
I see it a lot in discussions of cognitive science/artificial intelligence Interesting. I wasn't aware of this, but it is easy to see how this can happen.
There is perhaps an appearance of this sort of problem in cosmology, but I suspect that most physicists and cosmologists well understand the need to involve reality. Most excellently stated. And I certainly agree that there is an "appearance" to the outsider...
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Ben! Member (Idle past 1429 days) Posts: 1161 From: Hayward, CA Joined: |
Take the counting of objects as an example: no two objects are really completely, purely 100% identical, so there is only 1 of anything. We "fuzzy" the definition of "object" to include a set of extremely similar objects - idealize them - to apply the intellectual concept. I'm not so sure this is a valid objection. After all, if you're going to talk about "reality", then you have to also say that there's actually not any "1" thing at all; "objects" change all the time, whether it's their molecular motion or some atoms getting scrubbed off, etc. And if you really want to talk "reality" and abstraction... then talk about how "objects" are "identified" and "persist", even though the input signals we get from the world basically show us CHANGE. "Objectness" itself is an abstraction, an abstraction of the human mind. It fits right in with all other types of categorization. So I would say that, exactly in the same way that math is an abstraction of some "reality", so is our consciousness. We live an abstraction. And in the same way, I would say, don't mistake the abstraction for the "reality". Don't mean to push this off topic, just to point out that ... if you're going to go the philosophical route you are going, I think it's proper to "complete" the route. Ben
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RAZD Member (Idle past 1436 days) Posts: 20714 From: the other end of the sidewalk Joined: |
Gottlob Frege
Frege's Advances in Logic: Frege virtually founded the modern discipline of mathematical logic. He developed a system of conceptual notation (inspired by Leibniz's conception of a rational calculus), and though we no longer use his notation, his system constituted the first predicate calculus. Frege's second-order predicate calculus was based on the `function-argument' analysis of propositions and it freed logicians from the limitations of the `subject-predicate' analysis of Aristotelian logic. Frege's formal system made it possible for logicians to develop a strict definition of a proof. Unfortunately, Frege employed a principle (Basic Law V) in his later system (Grundgesetze) which turned out to be inconsistent. Despite the fact that a contradiction invalidated his system, Frege validly derived the Peano Axioms governing the natural numbers from a powerful and consistent principle now known as Hume's Principle (some philosophers have proposed that the derivation of the Peano Axioms from Hume's Principle should be called `Frege's Theorem'). Frege is most well-known among philosophers, however, for suggesting that the expressions of language have both a sense and a denotation (i.e., that at least two semantic relations are required to explain the significance of linguistic expressions). This seminal idea in the philosophy of language has inspired research in the field for over a century. Interesting. Thanks. we are limited in our ability to understand by our ability to understand RebelAAmerican.Zen[Deist
... to learn ... to think ... to live ... to laugh ... to share.
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RAZD Member (Idle past 1436 days) Posts: 20714 From: the other end of the sidewalk Joined: |
After all, if you're going to talk about "reality", then you have to also say that there's actually not any "1" thing at all; "objects" change all the time, whether it's their molecular motion or some atoms getting scrubbed off, etc. Or the subatomic level where particles change from one to another, yes, true enough, and that just makes the idealized concept of a set of objects just that much more fuzzy at any substantiative closer level of inspection. the mathematical concept of integers doesn't.
I would say, don't mistake the abstraction for the "reality". map. we are limited in our ability to understand by our ability to understand RebelAAmerican.Zen[Deist
... to learn ... to think ... to live ... to laugh ... to share.
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cavediver Member (Idle past 3674 days) Posts: 4129 From: UK Joined: |
Because math is a purely - and idealized - intellectual construction that does not exist outside the mind. This is a statement of one's philosophy of course; a philosophy with which many mathematicians would not agree.
Take the counting of objects as an example: no two objects are really completely, purely 100% identical, so there is only 1 of anything. At your length scale, you are approximately correct. But you are made up of fermions and bosons. These are identical... completely identical. This is why we have the Pauli exclusion principle, which is a good thing because we couln't exist without it!
We could argue about the concept of a plane as a surface, but let's cut to the quick and take {the still simple yet one level more complex concept of} the mobius strip: no such thing exists in reality. I'm not sure what you mean here. I have made hundreds of mobius strips. A mathematical space can be realised by far more than just a surface. Rotations of objects are better realisations, and rotations of fermions and bosons are even better. And these spaces are far far more bizarre than the good old mobius band. Mathematics is certainly idealised at the length scales with which you are familiar... I have no problem with that. But I am not referring to those length scales.
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RAZD Member (Idle past 1436 days) Posts: 20714 From: the other end of the sidewalk Joined: |
I'm not sure what you mean here. I have made hundreds of mobius strips. With edges, joints, and thickness. you might as well make a donut and call it a mobius strip.
made up of fermions and bosons. always the same ones? we are limited in our ability to understand by our ability to understand RebelAAmerican.Zen[Deist
... to learn ... to think ... to live ... to laugh ... to share.
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nwr Member Posts: 6412 From: Geneva, Illinois Joined: Member Rating: 5.3 |
Because math is a purely - and idealized - intellectual construction that does not exist outside the mind.
That's a strange thing to say. After all, our concept of "object" is an idealized intellectual construction that does not exist outside the mind.
Take the counting of objects as an example: no two objects are really completely, purely 100% identical, ...
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Ben! Member (Idle past 1429 days) Posts: 1161 From: Hayward, CA Joined: |
Gee nwr, thanks for saying what I said in 4 paragraphs, in 2 lines.
Back to the drawing board One of these days, I'll figure out this whole communication thing.
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cavediver Member (Idle past 3674 days) Posts: 4129 From: UK Joined: |
With edges, joints, and thickness You are confused in your concept of mathematical space. A mobius band is a statement about the topology of a space. My cellotaped piece of paper has the correct topology and is hence a mobius band. The thickness is immaterial. The point is that a mobius band need have no thickness to be a mobius band, in the same way that a 2-sphere has no interior. The surface of the earth is a 2-sphere, despite the earth having an interior.
always the same ones? How would I tell? They are identical. This is a pivotal point of quantum/particle physics. It is not an idealisation. The quantum theory of electrons falls apart if there is the slightest hint of even a theoretical possibility of being able to distinguish two electrons.
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robinrohan Inactive Member |
Maths is an entirely human construct I don't see this. When math conflicts with common sense, math always wins. Quantum philosophy. The universe is formal. Elegant. "And from water we made all living things."-- The Quran
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cavediver Member (Idle past 3674 days) Posts: 4129 From: UK Joined: |
Following on from my discussion with RAZD...
You have two electrons. Shake them up so they are equally likely to be spin up or spin down. What is the prob they are both spin up?What is the prob they are both spin down? What is the prob one is spin up and the other spin down?
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RAZD Member (Idle past 1436 days) Posts: 20714 From: the other end of the sidewalk Joined: |
It all comes down to perceptions of reality eh?
we are limited in our ability to understand by our ability to understand RebelAAmerican.Zen[Deist
... to learn ... to think ... to live ... to laugh ... to share.
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RAZD Member (Idle past 1436 days) Posts: 20714 From: the other end of the sidewalk Joined: |
My cellotaped piece of paper ... ... models the mathematical concept close enough to convey the idea, but it is not the concept. Topologically or otherwise.
The quantum theory of electrons falls apart if there is the slightest hint of even a theoretical possibility of being able to distinguish two electrons. And yet, when you look inside the subatomic particles are forever dancing and changing partners ... based on QT eh? And you seem to be confusing "theoretical" with "actual" possiblity of distinguishing, are you claiming that theory is more important? Isn't that just exactly the point here? If it WAS measured and DID invalidate QT what would happen to reality? we are limited in our ability to understand by our ability to understand RebelAAmerican.Zen[Deist
... to learn ... to think ... to live ... to laugh ... to share.
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