# Mathematics in physics?

1. Jun 5, 2005

### pivoxa15

What I really like to know is 'Does mathematics fully or truly represent physical reality?'

For example, when we use mathematics in physical problems, are we dealing with 'what is really going on in reality'? Or are we simply approximating (in many cases a really good approximation but still an approximation) reality?

The word reality in this context is 'what is truly or really happening (regardless of human existence)'.

Last edited: Jun 5, 2005
2. Jun 6, 2005

### whozum

This sounds more like a philosophical question, you might have better luck in the philosophy forum.
My opinion is that math accurately describes physical phenomena but what baffles me is that some of those phenomena are simple enough to be described by some basic equations.

3. Jun 6, 2005

### pivoxa15

Thanks for your advice whozum. I have posted a new thread in 'Philosophy of Science and Mathematics'.

4. Jun 6, 2005

### arildno

Counter-question:
Does it matter?

5. Jun 6, 2005

### metrictensor

I don't agree with your definition of reality. If there is no mind there would be no reality. We not placed in the world like water is placed in a glass. We are a participant in the world. Even quantum mechanics (to some extent) supports this view.

6. Jun 6, 2005

### neurocomp2003

depends: at the truly fundamental(if it exists and if we discover it...i wonder if it is observable) the answer is yes? After all math is all about fundamental things and counting them. But currently one can argue both ways but math is a model for us to comprehend...because its easier to make a set of equations then to observer a multimillionobject system.

7. Jun 6, 2005

### εllipse

Here is a quote that I think sums it up well.

- Ray D'Inverno, Introducing Einstein's Relativity, Section 2.1

Last edited: Jun 6, 2005
8. Jun 6, 2005

### X-43D

Actually we use calculus and geometry. Almost all of physics is based on it. In any case, there is a chance that nature is simply not geometric.

9. Jun 6, 2005

### metrictensor

This is an excellent quote. In light of this and many other similar examples some scientists still hold on to the idea of TOE's and such.

10. Jun 6, 2005

### metrictensor

Nature is not geometric and it is not non-geometric. Scientific models are conceptualizations of reality that in some cases do model nature very well. But they do not describe nature as it objectively is.

To quote Heisenberg "Since the measuring device has been constructed by the observer ... we have to remember that what we observe is not nature itself but nature exposed to our method of questioning."

11. Jun 6, 2005

### X-43D

I agree. Nature doesn't know calculus, almost surely.

Newton invented calculus in 1665 to describe the motion of celestial bodies but he didn't know what the source of motion was. Also, he didn't know what mass, weight and gravity are. In fact, he never pretended to.

In 1900, the discovery of radiation by chemists made it clear that nature is only about particles, their intrinsic properties and the interactions betweem them.

The mathematics we are using was originally invented for engineering, design, industry, business and economics.

Last edited: Jun 6, 2005
12. Jun 6, 2005

### whozum

Be very careful with what you say. I strongly disagree with your first statement. Also, calculus, among math, is a tool used to explain physics, it is only discovered, never invented.

On this note, I want to ask a question:

Is physics discovered as the math explaining it is developed, or is the math born as new physical phenomena need an explanation?

13. Jun 7, 2005

### metrictensor

This is a highly contentious issue. Consider the diagonal of a square. Assume we assume the existence of a square. An emergent property is that the diagonal is sqrt(2)*side length. But this truth depends upon the existence of a square and therefore could not have existed before we posited squares. So is the property discovered or invented?

14. Jun 7, 2005

### pivoxa15

Thanks for your replys.

metrictensor, by saying "If there is no mind there would be no reality" do you mean that the period before the first human set foot on earth, reality (things or events in this world) did not exist?

Countless evidence suggests that things did happen in this universe before humans existed.

With regards to the quote by Ray D'Inverno. Is he suggesting that whenever we try to describe the world, we must use models but no model is ever 100% accurate to what actually happens in reality? Hence we can only use the best one. The one that approximates reality best.

I have two (basic - I am only doing first year University physics) ideas and their philosophical consequences about using mathematics in physics.

The real number line with an infinite number of numbers is defined in mathematics. We use these lines in the cartesian plane. We can set up a relationship between two or more variables and graph them on this plance. i.e. D(t)=2t
Distance as a function of time. In reality, this object may behave to this relationship really closely. But this function assumes that time and matter are continous. However, is it in reality?
If it isn't than it means that the foundations of mathematics, the real number line cannot be applied in physics (although it could if all we are after is an approximation. In this case, it would be a really good one)

The other thing, related to the above is the idea of an instanteous rate of change. I think that an instanteous rate does not exist in reality. i.e. take a photograph of a moving car. You know where it is at one instant but there is no way you could tell its velocity.
There is also a problem with the mathematics of instaneous rate of change. They use the notion of a limit. It is approaching a value but will never equal the value it approaches. Hence the instaneous rate of change does not exist mathematically or physically. Although a rate of change so close that we can approximate it as instaneous (again, this approximation is so good that it is as good as our imagination (mathematically) - alhtough if time and matter were discrete than things will be stricter-since we can only use two smallest units of time and matter for our calculation)

So I think that mathematics does not represent physical reality as most of you suggest.

What do you think?

15. Jun 7, 2005

### metrictensor

There were animals before humans. It is the idea that there would be not universe if there were no being to inhabit it. It doen's mean there have to be beings at each particular place.

Last edited: Jun 7, 2005
16. Jun 7, 2005

### X-43D

You are right. Calculus and geometry does not represent physical reality.

The problem with calculus is that it fails to explain the basic particle interactions which turn mass into weight.

17. Jun 7, 2005

### selfAdjoint

Staff Emeritus
This of course was a question that interested the ancient Greeks. It is precisely what Zeno's paradoxes are about. There is a modern program to do physics without continuous change; google on "digital physics" or read Kurzweil's A New Kind of Science

Funny, my car has a speedometer. Why don't you look into how a speedometer is able to make that needle move based on velocity?

This is just false, and shows you haven't properly internalized the genuine limit concept. The limit exists because the real line is complete. Of course you suggest denying that above, but your argument here is assuming it. The limit isn't defined by the approximating sequence. And there is no time based relation either; we don't have first a1, then a2 a moment later, then a3, and so on. This was Zeno's misunderstanding. The sequence is defined all together all at once as part of the real line geometry.

You have two issues here. One is a conjecture that physics - the real world - may not be complete, that not every bounded set of points will have a limit point. This is possible but so far "there is no need for that hypothesis". The other is that even in a complete and continuous world, the limit is not well defined, and this is wrong.

Many beginning physics students have ideas like this, especially if they are taking beginning calculus at the same time. Wait till your understanding is deeper and then reexamine your ideas.

18. Jun 9, 2005

### pivoxa15

On p502, in the appendix section in "A COURSE ON PURE MATHEMATICS" by G.H Hardy

He wrote “The infinite of analysis is a ‘limiting’ and not an ‘actual’ infinite. The symbol 'infinity' has throughout this book been regarded as an ‘incomplete symbol’, a symbol to which no independent meaning has been attached, though one has been attached to certain phrases containing it.”

So Hardy is suggesting, for example that the summation of 1/2+1/8+1/16+1/32...1/n as n limits to infinity will become extremely close to 1 but it will never actualise or become exactly 1.

Likewise with the calculus, the notion of a limit only allows a number to approach a certain value but never ever becoming or equaling it. Hence an 'instanteous' derivative is not possible, mathematically.

So on this matter I totally agree with Hardy. But selfAdjoint you seem to suggest otherwise?

19. Jun 9, 2005

### mathwonk

i think you fail to understand either selfadjoint, or hardy. the passage you quote does not say what you think it does.

the point with the limit of an infinite sequence a1,a2,...is not that some one of the ai's should get to the limit, but that YOU should get there.

e.g. the limit of the decreasing sequence 1/1, 1/2, 1/3,.... may be defined as "the largest number which is not larger than any number in the sequence". If you think hard enough about this definition (and accept the archimedean axiom) you may arrive at the understanding that the number described must be 0.

It has nothing to do with whether any element of the sequence is ever zero.

similarly, the "sum" of the series 1/2 + 1/4 + 1/8 +.... is defined as the smallest number not smaller than the sum of any finite number of terms in the series. If you understand that this number is 1, you have correctly evaluated the sum. Of course no finite part of the series ever gets there, so what? that is not the definition of an infinite sum.

As to Hardy, you seem confused by his attempt to explain his use of the word "infinity".

Last edited: Jun 9, 2005
20. Jun 10, 2005

### X-43D

I think calculus fails when it comes to explain motion and gravitation.

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