Is math invented or discovered?
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25-10-2016, 05:35 AM
RE: Is math invented or discovered?
(25-10-2016 04:23 AM)Mathilda Wrote:  I'm not a physicist or mathematician but I read in Einstein's biography that he had to develop a form of non-Euclidean geometry to describe his general theory of relativity, the Einstein tensor.

That's not to argue that the universe doesn't work in a particular way, but we have to develop a language to describe it. Maybe there was a better way to describe it which someone else might yet devise, who knows?

I can speak as a computer scientist though, CS being an applied form of Maths and logic. I research Artificial Intelligence. I create strong AI. The only natural form that we have of this involves the use of neurons which work in a particularl way. There are good reasons why nature used neurons, but it's very difficult to understand how networks of neurons function and why they work the way the way they do. And consequently, this also means that it's very difficult to actually work with artificial biologically plausible neural networks and get them to do something different, to engineer them.

Instead I describe them using a dynamical systems approach. I did this out of necessity. I devised my own mathematical model that describes how the networks function in a way that allows me to reason about them far more simply and successfully. It abstracts away all the unnecessary detail. In fact I've now spent the last 6 years developing an implementation based on the dynamical systems concept that is more useable than biologically plausible neural networks. Both dynamical systems and neural networks can do exactly the same thing, and just as well as each other it seems, but the former is easier to understand and consequently to play with.

So no sign of Assimov's "positronic brain" yet then? Smile

Though not sure, using the modern definition, if I fancy heads full of anti-matter wandering around!

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25-10-2016, 05:53 AM
RE: Is math invented or discovered?
(25-10-2016 05:35 AM)Gloucester Wrote:  So no sign of Assimov's "positronic brain" yet then? Smile

You could say that!

Although it's relatively easy to evolve a neural network that's really big and complicated and does something relatively, you don't then know how it works and can't do anything with it.

I spent the best part of a year trying to get one three layer neural network that I fully understood to feed into two other copies of itself.

I had this idea of using neural networks like integrated circuits, try to understand how each component works individually and then piece them together. I gave up in the end. Nothing ever works in AI the way you hope it will.




(25-10-2016 05:35 AM)Gloucester Wrote:  Though not sure, using the modern definition, if I fancy heads full of anti-matter wandering around!

There'd be an explosion if it came into contact with matter. Bad news for people who keep their head up their own arse.
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25-10-2016, 06:38 AM
RE: Is math invented or discovered?
(25-10-2016 01:27 AM)Mathilda Wrote:  And what of all the problems that can't be adequately solvdd using any form of known Maths, logic or language? We then create a new language or model to work with that can describe the problem and be manipulated more concisely. It happens all the time.

Saying that Maths exist outside of a concept has no explanatory power. What exactly are these Mathematical laws made from? How do they have the universal effect that they do?

I don't disagree that we can develop mathematical systems for the purpose of solving problems, I just pointed out that it is not always the case. Furthermore, the fact that it is not always the case raises the question of how does something of our own invention appear so intrinsically woven into reality. We create many useful tools to solve problems, but only mathematical concepts (unless I'm mistaken) have been found in nature as well. I will point to the Fibonacci sequence as the obvious example. Most children have at some point encountered the Fibonacci sequence because it is so easily produced, but they have observed it without knowing that the Fibonacci sequence appears in the structure of plants. No problem to solve here, it was just observed.

Asking what mathematical laws are made from is an interesting question. What do you mean by "made from"? Are you asking what elementary particles does it consist of? What space does it take up and at what time? If I were to properly answer your questions, I would say that it is because mathematics is the fundamental nature of the universe. That is, it isn't composed of physical elements, rather the physical elements are actually mathematical constructions. Our mathematical laws have a universal effect because they model the true mathematical nature of the universe. And if you continue to ask "what is that made of", then you might as well ask "what is energy made of" or whatever you think seems to be the fundamental properties of the universe.
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25-10-2016, 06:54 AM
RE: Is math invented or discovered?
(24-10-2016 09:59 PM)Fireball Wrote:  Mathematics fits the observations because we make up equations that do that. That's one of the points of using mathematics- to make it describe what is happening!

I think the point of interest is not how mathematics is able to explain observations, because we can build systems to solve problems (and I do find that story quite humorousSmile). Rather, it's how mathematics is able to solve problems without even being built to do so. Sometimes it doesn't even solve problems that we need to solve. Again, we simply observed that the Fibonacci series exists in the structure of plants. No problem there for us to solve, it just is.
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25-10-2016, 07:26 AM
RE: Is math invented or discovered?
(25-10-2016 01:27 AM)Mathilda Wrote:  
(24-10-2016 09:51 PM)unknowndevil666 Wrote:  Like I've said, we create our own mathematical systems like geometry and calculus. But it is in the unexpected use of the systems in understanding reality that implies there is more meaning to be found here than simply calling them useful tools, like a hammer or saw. When I speak of mathematics, I am talking of the order that allows any of our mathematical systems to work. You say that we invent languages to solve new problems, but often problems are solved by languages thought to be purely abstract. The appearance of mathematical concepts in nature suggests that what we call mathematics exists outside of our own invention.

And what of all the problems that can't be adequately solvdd using any form of known Maths, logic or language? We then create a new language or model to work with that can describe the problem and be manipulated more concisely. It happens all the time.

Saying that Maths exist outside of a concept has no explanatory power. What exactly are these Mathematical laws made from? How do they have the universal effect that they do?

Yep. Math attempts to reduce the world into a form that we can use to understand it and gain insight about it. It is a tool. Saying that the world is mathematical is like saying lumber is sawy, because a saw does such a fine job of slicing through it.

We have to remember that what we observe is not nature herself, but nature exposed to our method of questioning ~ Werner Heisenberg
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25-10-2016, 07:26 AM
RE: Is math invented or discovered?
(25-10-2016 06:54 AM)unknowndevil666 Wrote:  I think the point of interest is not how mathematics is able to explain observations, because we can build systems to solve problems (and I do find that story quite humorousSmile). Rather, it's how mathematics is able to solve problems without even being built to do so. Sometimes it doesn't even solve problems that we need to solve. Again, we simply observed that the Fibonacci series exists in the structure of plants. No problem there for us to solve, it just is.

Yes it is, but so what?
Gravity is an inverse square ratio, but so what?
Pi can be calculated to any degree of significance with a Taylor series, but so what?
Why does it seem significant to you?

You seem to be sliding into "fine-tuning" territory. Consider

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25-10-2016, 07:45 AM
RE: Is math invented or discovered?
Maths is discovered, mathematical patterns, more so geometrical patterns are without shortage in the cosmos, which have been there before humans ever existed. What's interesting some of these patterns are necessary for us to have life as it is.
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25-10-2016, 07:54 AM
RE: Is math invented or discovered?
(25-10-2016 07:45 AM)Echo Wrote:  Maths is discovered, mathematical patterns, more so geometrical patterns are without shortage in the cosmos, which have been there before humans ever existed. What's interesting some of these patterns are necessary for us to have life as it is.

Yup, can't quite imagine a solar system without circles and ellipses! Big Grin

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26-10-2016, 03:39 PM
RE: Is math invented or discovered?
(20-10-2016 07:24 PM)Bucky Ball Wrote:  Math is learned.
Humans are pattern seeking monkeys.

That doesn't say anything about whether math is discovered or invented. I can learn about the water cycle (discovered) and I can learn about car engines (invented). Your second comment can also go either way. Does "pattern seeking" mean that nature is filled with patterns that we can find or that we find patterns even where there are none, or something else? I feel like you're saying it's invented though.
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27-10-2016, 12:32 AM
RE: Is math invented or discovered?
(25-10-2016 07:26 AM)Chas Wrote:  Yes it is, but so what?
Gravity is an inverse square ratio, but so what?
Pi can be calculated to any degree of significance with a Taylor series, but so what?
Why does it seem significant to you?

It's significant because if an "inverse square ratio" is just the product of man-made mathematical convention, it shouldn't exist outside of any man-made problem unless there is some underlying function that extends beyond the specific case. Take language for example:

Any particular language (English, Chinese, etc.) is a tool, a man-made invention. We don't observe letters or symbols of any language in nature. However, there is a underlying, fundamental property of all languages: communication. Communication is not simply a man-made tool. It is used by humans, but we didn't originate it. We also see communication in nature, among other animals. Math is the same way.

Particular systems of mathematics, such as geometry, calculus, number theory, can be viewed as the different languages. We have, conventionally, specified these groups. Another convention, we don't see the symbols that we use (1, *, %) in nature. However, like spoken languages, there is an underlying, fundamental property of all mathematical languages. What is it?

I'll call it "relation" for right now. Mathematics shows how things relate to each other. Take any law of physics, F=G(M*m)/r^2 for instance. Newton didn't just create this formula of his own convention, he discovered this relationship between masses and put it into the syntax of our mathematical language, the same way we can communicate real concepts with man-made words in our English language. And "relation" exists in every field of mathematics, it arises from operation on truth values, logically proving theorems from axioms, etc. One might even say that "relation" doesn't just exist in math, but is the math.

What this all means is that if the "inverse square ratio" was something created by humans for specific problems, it should stay within the realm of those problems. The fact that we've found this is embedded in universal law suggests it's a "relation" that exists outside of our own convention.

You also use pi as an example. The significance of pi being able to be calculated in a Taylor series is that it was originally found by using polygons and circles. If pi was created as a tool to solve geometric problems, then what the hell is it doing in calculus or the Gaussian distribution?? Our languages of mathematics can define pi using C/d, or the integral of a half-circle function, or Euler's formula (like how one can use different languages to communicate anger) but pi means something more than just the way we use it (like how the language in which we communicate anger is not the actual communication itself).

So yes, we have different methods of mathematics (methods of communication, i.e. language) that we create and use as tools, but that isn't the essence of mathematics itself.

Anyway, I have to go to bed. I'm ending it here.
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