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Math v physics (split from Direction of time)

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On 8/30/2019 at 1:26 PM, swansont said:

math is not constrained by physical laws

This sounds like coming from ancient Greek philosophy.What are the arguments that support this statement? And I wonder, are there arguments against this? I mean, scientific arguments. Not philosophical ones. For sure arguments can not come out from physics because if so, it would be a counter-argument.

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1 hour ago, michel123456 said:

This sounds like coming from ancient Greek philosophy.What are the arguments that support this statement? And I wonder, are there arguments against this? I mean, scientific arguments. Not philosophical ones. For sure arguments can not come out from physics because if so, it would be a counter-argument.

I don't understand how physical laws could be relevant to mathematics. And I can't see what sort of scientific arguments could be made either way. 

There are various philosophical arguments that can be made about mathematics (is it discovered or invented, for example) but the idea that it is somehow dependent on physical laws is a new one to me. Do you have any examples of how it might be?

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1 hour ago, Strange said:

I don't understand how physical laws could be relevant to mathematics. And I can't see what sort of scientific arguments could be made either way. 

 

Read this

https://books.google.co.uk/books/about/The_Mathematical_Mechanic.html?id=lW5vQK6Tcu8C&printsec=frontcover&source=kp_read_button&redir_esc=y#v=onepage&q&f=false

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3 hours ago, michel123456 said:

This sounds like coming from ancient Greek philosophy.What are the arguments that support this statement?  

Math requires that it be internally consistent. It does not require that it be compared to nature's behavior. That's why math can prove things.

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2 hours ago, Strange said:

I don't understand how physical laws could be relevant to mathematics. And I can't see what sort of scientific arguments could be made either way. 

There are various philosophical arguments that can be made about mathematics (is it discovered or invented, for example) but the idea that it is somehow dependent on physical laws is a new one to me. Do you have any examples of how it might be?

I understand James Wheeler's One electron universe as showing that the concept of unit (the number 1) is directly linked to time. If a unique thing (unit) can be at different point of spaces at the same time it fails to be one. And if the concept of 1 is under scrunity, I suppose that maths should be under scrunity too. Maybe is that the reason of our difficulty to understand entanglement, when 2 particles separated by space behave as if it was the same and one particle (1 unit composed of 2 separated intervals on the number line).

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1 hour ago, michel123456 said:

If a unique thing (unit) can be at different point of spaces at the same time it fails to be one. And if the concept of 1 is under scrunity, I suppose that maths should be under scrunity too.

No. The concept of '1' is already a bit older... Maths creates axiomatic definitions and propositions, and based on logic shows what propositions can be derived from these. Where it might be true that maths historically was derived from practical use (counting coins, weighing gold or potatoes, measuring distances, etc), it later became more and more theoretical, creating its own 'mathematical objects', and study their properties. The interesting thing is that some mathematics turned out to useful in natural sciences: things like group theory, matrix algebra and differential geometry already existed before people like Heisenberg or Einstein started to use them. And both seemed to have a hard time in the beginning to learn these disciplines.

Point is: sometimes one finds things in reality, that fit to some mathematical structure, and then one can immediately start to reap the fruits. String theory is an example in the opposite direction: new mathematical methods had to be thought out, which is the reason that at least one physicist (Witten) won the Fields medal.

Edited by Eise

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2 hours ago, studiot said:

That appears (from the introduction) to say that mathematics exists independently of physics, although you can use physical/mechanical techniques to tackle some mathematical problems. This is entirely consistent with what I was saying. Unless one were to argue that the only valid mathematics is that which can be tackled by mechanical means.

 

1 hour ago, michel123456 said:

I understand James Wheeler's One electron universe as showing that the concept of unit (the number 1) is directly linked to time.

It is about electrons, not the abstract concept of the number 1.

Eise (and swansont) has said anything else I might want to say about (far better than I could!)

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52 minutes ago, Strange said:

That appears (from the introduction) to say that mathematics exists independently of physics, although you can use physical/mechanical techniques to tackle some mathematical problems. This is entirely consistent with what I was saying. Unless one were to argue that the only valid mathematics is that which can be tackled by mechanical means.

Your words were

Quote

I don't understand how .....................

So I offered you a source who did exactly that.

 

Detailed discussion of the relationship between Maths and Physics is complicated and extensive and off topic in a subject about the direction of time.

So I was brief.

 

 

Similarly here is a note to Michel

6 hours ago, michel123456 said:

This sounds like coming from ancient Greek philosophy.What are the arguments that support this statement?

The Ancient Greeks had a Philosophy of "ideals" which linked the physical world to the theoretical world.
This reappears in modern algebra a generalisation called the theory of ideals.

 

Edited by studiot

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56 minutes ago, Strange said:

That appears (from the introduction) to say that mathematics exists independently of physics, although you can use physical/mechanical techniques to tackle some mathematical problems. 

Yes. And since physics follows mathematically-expressed laws, that's should be unsurprising.

I recently bought a Galton board which drops tiny balls into a stack of pins, laid out in a triangular grid, such that there's nominally a 50-50 chance of going right or left with each layer. The balls collect in a gaussian pattern (plus noise)

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1 hour ago, studiot said:

Your words were

Quote

I don't understand how .....................

So I offered you a source who did exactly that.

Context is everything.

I was responding to michel123456 who seemed to think the statement "math is not constrained by physical laws" was implausible. 

1 hour ago, studiot said:

Detailed discussion of the relationship between Maths and Physics is complicated and extensive and off topic in a subject about the direction of time.

I agree. But I would like to hear michel123456's views on this. Maybe it should be split off to a separate thread.

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55 minutes ago, Strange said:

Maybe it should be split off to a separate thread.

I agree

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6 hours ago, michel123456 said:

I understand James Wheeler's One electron universe as showing that the concept of unit (the number 1) is directly linked to time. If a unique thing (unit) can be at different point of spaces at the same time it fails to be one. And if the concept of 1 is under scrunity, I suppose that maths should be under scrunity too. Maybe is that the reason of our difficulty to understand entanglement, when 2 particles separated by space behave as if it was the same and one particle (1 unit composed of 2 separated intervals on the number line).

Related to that, I did used to work with someone who insisted that you couldn't really define numbers because if you take "1 pebble" as an example, there is no clear boundary to what is pebble and what isn't. There will be a layer of dirt and moisture on the outside and if you wipe that off, you will also remove part of the pebble itself. (Perhaps not surprisingly, he was an engineer!)

Now while I agree that it can be hard (if you get really picky) to draw the line between physical objects, the [modern] idea of numbers is not based counting pebbles. Instead, the symbol "1" is used to define an abstract concept which can be defined in terms of axioms and rules.

Mathematics is just a set of rules for manipulating symbols that we give particular meanings to. So the number 1 doesn't relate to pebbles or electrons. It just has to behave in particular ways.

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9 hours ago, michel123456 said:

This sounds like coming from ancient Greek philosophy.What are the arguments that support this statement? And I wonder, are there arguments against this? I mean, scientific arguments. Not philosophical ones. For sure arguments can not come out from physics because if so, it would be a counter-argument.

Physical laws have certain properties, and you can have maths that have the same properties, but you can also have other maths that don't, including abstract mathematics that aren't based on physical things.

For example, you can mathematically take apart an orange and rearrange it into two oranges identical to the original. https://en.wikipedia.org/wiki/Banach–Tarski_paradox The maths aren't restricted by the physical law of conservation of mass.

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6 minutes ago, md65536 said:

For example, you can mathematically take apart an orange and rearrange it into two oranges identical to the original. https://en.wikipedia.org/wiki/Banach–Tarski_paradox The maths aren't restricted by the physical law of conservation of mass.

Great example +1.

49 minutes ago, Strange said:

Related to that, I did used to work with someone who insisted that you couldn't really define numbers because if you tack "1 pebble" as an example, there is no clear boundary to what is pebble and what isn't. There will be a layer of dirt and moisture on the outside and if you wipe that off, you will also remove part of the pebble itself. (Perhaps not surprisingly, he was an engineer!)

Interesting example.

Sort of harps back the the Ancient Greeks' idea that although say a sugar cube must contain the points of a perfect circle, no perfect circle exists in Nature.
Similarly most of their maths was plane geometry, although no plane exists in Nature.
Both are 'ideals' in their sense.

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14 hours ago, md65536 said:

Physical laws have certain properties, and you can have maths that have the same properties, but you can also have other maths that don't, including abstract mathematics that aren't based on physical things.

For example, you can mathematically take apart an orange and rearrange it into two oranges identical to the original. https://en.wikipedia.org/wiki/Banach–Tarski_paradox The maths aren't restricted by the physical law of conservation of mass.

Nice argument. +1

So if I understand clearly the above, the reverse operation is mathematically doable. That is 1 orange + 1 orange = 1 orange. Isn't it?

And in this case, doesn't it raise the question (again) of what is 1?

18 hours ago, swansont said:

Yes. And since physics follows mathematically-expressed laws, that's should be unsurprising.

I recently bought a Galton board which drops tiny balls into a stack of pins, laid out in a triangular grid, such that there's nominally a 50-50 chance of going right or left with each layer. The balls collect in a gaussian pattern (plus noise)

Nice! The Galton board is exactly the kind of example that makes me think that it is not a coincidence that maths correspond to physics. It may be that maths are so abstract that they relate to the abstract background of the real world. Or it may be that maths have been created on the basis of the real physical world. IMHO the latter sounds more realistic.

 

Quote

Could this be related to Mandelbrot's fractals?

Edited by michel123456

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52 minutes ago, michel123456 said:

So if I understand clearly the above, the reverse operation is mathematically doable. That is 1 orange + 1 orange = 1 orange. Isn't it?

Good question. I  might assume it is reversible, but when dealing with things that are so counterintuitive...

52 minutes ago, michel123456 said:

And in this case, doesn't it raise the question (again) of what is 1?

I don't think so. Because this is dealing with one orange, not the abstract concept of 1.

And it is not even a physical orange. One can't do this with real objects; just mathematical ones.

52 minutes ago, michel123456 said:

Nice! The Galton board is exactly the kind of example that makes me think that it is not a coincidence that maths correspond to physics. It may be that maths are so abstract that they relate to the abstract background of the real world. Or it may be that maths have been created on the basis of the real physical world. IMHO the latter sounds more realistic.

It is clear that maths and physics are related. And that seems to go deeper than just the fact that we use mathematics to describe the world. Or even the fact that we can use mathematics to describe the world. It does seem as if the physical world is actually defined by mathematics (rather than the other way round).

There is some interesting work by the mathematical physicist, Cohl Furey, on the idea that quantum theory can be derived from purely mathematical principles. I linked to a couple of articles on this a while ago: https://www.scienceforums.net/topic/116080-mathematics-physics-and-theory-of-everything/

Someone else posted a video of one of her lectures on the subject. I rarely recommend videos, but these are worth watching.

I suppose this is a more advanced version of the Greek's view of the universe being a mathematical construct (bringing us back, nicely, to your opening post).

 

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2 hours ago, michel123456 said:

 Nice! The Galton board is exactly the kind of example that makes me think that it is not a coincidence that maths correspond to physics. It may be that maths are so abstract that they relate to the abstract background of the real world. Or it may be that maths have been created on the basis of the real physical world. IMHO the latter sounds more realistic.

Corresponds is not nearly the same thing. And we know examples of maths having been created that had nothing to do with the physical world, because applications came after the math. There are undoubtedly maths out there that are not used in science.

Science, OTOH is a description of the behavior of world around us, and is based on mathematical relationships. The universe follows rules, and therefore science is based on maths.

But if you drew a Venn diagram (do these exist in nature?) scientific equations/relationships would be a wholly-contained subset of "Maths that exist" but would not be the entire set.

 

 

 

1 hour ago, Strange said:

 It is clear that maths and physics are related. And that seems to go deeper than just the fact that we use mathematics to describe the world. Or even the fact that we can use mathematics to describe the world. It does seem as if the physical world is actually defined by mathematics (rather than the other way round).

There's a quote by someone whose name currently escapes me about how it's amazing that we can describe the world using mathematics.

———

One example to show that math isn't dictated by nature is that we can come up with equations that are self-consistent, but don't describe how nature behaves. Newtonian/Galilean physics for example, is only an approximation, and fails at high speeds. e.g. the limitation of traveling at c is one the universe places on us, but not mathematics.  

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31 minutes ago, swansont said:

But if you drew a Venn diagram (do these exist in nature?) scientific equations/relationships would be a wholly-contained subset of "Maths that exist" but would not be the entire set.

The converse is also true.

There are parts of Physics (and other scientific disciplines) that are not in Maths.

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3 hours ago, swansont said:

There's a quote by someone whose name currently escapes me about how it's amazing that we can describe the world using mathematics.

 

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is the title of an article published in 1960 by the physicist Eugene Wigner.[1] 

https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences

And Max Tegmark has written a book called "Our Mathematical Universe" where he suggests the universe is "made of math."

https://www.scientificamerican.com/article/is-the-universe-made-of-math-excerpt/

(I have not read either of these.)

3 hours ago, swansont said:

But if you drew a Venn diagram (do these exist in nature?) scientific equations/relationships would be a wholly-contained subset of "Maths that exist" but would not be the entire set.

An interesting question is whether "maths that exist" is a superset of "maths we know", or are they the same thing.

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7 minutes ago, Strange said:

 An interesting question is whether "maths that exist" is a superset of "maths we know", or are they the same thing.

 

Since there are undoubtedly new maths that will be revealed at some point in the future, I'd have to say they aren't the same thing.

(To the extent that this is about whether maths are invented or discovered, as far as I am concerned that should read "uninteresting")

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On that point there is some new representations that has been hitting papers with regards to Dirac notation. Ie statements with more than one ket and bra statement.

Edited by Mordred

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2 hours ago, swansont said:

(To the extent that this is about whether maths are invented or discovered, as far as I am concerned that should read "uninteresting")

:) 

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12 hours ago, michel123456 said:

So if I understand clearly the above, the reverse operation is mathematically doable. That is 1 orange + 1 orange = 1 orange. Isn't it?

And in this case, doesn't it raise the question (again) of what is 1?

I think it's reversible, but it's not 1 + 1 = 1, because adding oranges together conserves oranges. The cloning operation isn't adding. Same with adding finite natural numbers (quantity is conserved), which have properties corresponding to everyday objects. Adding volumes of water conserves volume, but adding it to a black hole doesn't. Different mathematical objects have different properties and correspond to different physical things (some not at all).

In this case it's not about "1", but really about infinity, because decomposing the orange involves splitting it into an infinite number of points, and you can add two infinities together and end up with one infinity. So even natural numbers, which have properties I would guess are based on our understanding of the physical world, have properties not necessarily restricted by physical laws.

Edited by md65536

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1 hour ago, md65536 said:

I think it's reversible, but it's not 1 + 1 = 1,

 

I suppose that depends upon your definition of the operator +

 

I agree it can be reversible for some mathematical objects since the reverse involves superposition.

Two or more congruent mathematical objects such as triangles or balls can be superposed to end up with one mathematical copy of the object.

Physical objects present more of a difficulty as most times two physical objects cannot occupy the same space.

But what about two part glasses of water?

Pour one into the other and you have one part glass of water.

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11 hours ago, studiot said:

Physical objects present more of a difficulty as most times two physical objects cannot occupy the same space.

 

"Two physical objects cannot occupy the same space" at the same time.

They can occupy the same space at different times.

And that makes me think that maybe (maybe) the duplication of the Banach-Torski paradox is the result of not including time in the equation.

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