# Mathematics in Physics

1. Jul 28, 2008

### Euan

I love and am absolutely fascinated by both mathematics and physics but have no training of any worth in either. As such I should be grateful if anyone answering any of the three questions below avoids (if at all possible) terminology or language that requires such training. Thank you very much for your consideration.

My three questions are:

1. Which area (or areas) of mathematics is (are) basic in the sense that all other areas can be derived from it (them)? Or, alternatively, if this question is itself the ‘wrong’ one to ask, what is the ‘correct’ question along these same lines?

2. My understanding is that the two pillars on which physics is based are experiment and mathematics. I can see the value of experiments but why is mathematics (at the most fundamental level) so important in physics?

3. Given 2, why is it that, time after time, developments in mathematics that start out as ‘pure’ turn out to be key to providing insights into developments in physics?

Euan

Note to moderators. I don’t want to double post but should I also post this on one of the physics forums?

2. Jul 28, 2008

### Staff: Mentor

To some extent that's just like asking which organ is basic - is it kidneys, liver, brain, lungs or heart? You need them all.

Describes and let's you make quantitative predictions.

You are not the first one to ask such questions, see for example https://www.physicsforums.com/showthread.php?t=237609 - and don't expect any definitive answers

3. Jul 28, 2008

### spideyunlimit

2. you need maths to describe relations and functions between various phenomenon. ( Without which, physics is incomplete.)

4. Jul 28, 2008

### Euan

Borek

Euan

5. Jul 28, 2008

### Euan

Spideyunlimit

May I push you a little further on this and turn it around? Could one argue (from your reply) that physics studies phenomena that are best described by mathematical relations and functions (as opposed to, say, being described by a language such as English)? Indeed, might this be a definition of physics?

And, taking that idea further, phenomena that are not best described by mathematics – an emotional reaction to music, say – will not therefore form part of the subject matter of physics.

Euan

6. Jul 29, 2008

### robertm

Hello Euan, and welcome to PF,

Would you not agree, that if mathematics fails to properly explain a phenomenon, then the problem is more likely inherent to the language, and not the phenomenon? Assuming otherwise, would be drifting into irrationality, or the idea of the supernatural.

I would also argue that mathematics can most certainly describe the numinous and the transcendent; it simply is that the descriptions seem to cheapen the personal experience or seem to be inadequate so one is opt to dismiss as not applicable; when in fact electro-chemical reactions in one's brain is about as mathematical as it can get.

Physics needs mathematics. It is the only unambiguous language that we have been able to invent, the one that is most logical.

7. Jul 30, 2008

### Pythagorean

To expand on this, it's basically a higher-resolution of "none, a little bit, some, and a lot".
other qualitative words, like "low, medium high" and "cold, moderate, hot" can actually still be considered quantitative (if they are properly defined) but they have a very low resolution. As we increase the resolution, coming up with descriptive words to describe all the points between cold and moderate becomes tiresome. So we use much simpler symbols: numbers.

More incredibly, we begin to find relationships between these numbers (if you have two sets of two, you have four... 2x2 is 4...).

This is very basic mathematics though; for a physicist that studies only the bare math, a large majority of the math relates to functions. Functions are basically a system that takes a number in and spits a result out. Every time we put the same number in, we'll gate the same result back, and only one result.

We can use functions to model things in physics. For instance, gravity follows the inverse square law. The inverse square is a fancy way for a function that looks like this:

$$f(r) = \frac{1}{r^2}$$

we would input a number (replacing r with that number) and find that as the number "r" gets bigger and bigger, the function, f(r) gets smaller and smaller. As I said before, this relates to gravity (and many other things in nature), where r is your distance from Earth. As you get farther and farther away from Earth (r increases), it's gravity has less and less effect on you (f(r) decreases).

8. Jul 30, 2008

### Pythagorean

My opinion on 3:

There's plenty of mathematics that is useless to physics too, but I think the main reason is simply because mathematics is cheaper and easier to develop than technology that pushes the boundaries of physics, and therefore develops at a much faster rate. (Said clearly, mathematics develops faster than physics)

Even in technology and physics, some of the best discoveries were the results of mistakes (from plastic to Bosons).

I think theres enough of us out there, spending enough time, and enough prior research at this point (to lend to learned academic intuition) that we can't help but have so many coincidences.

9. Jul 30, 2008

### Euan

Robertm

Would you not agree, that if mathematics fails to properly explain a phenomenon, then the problem is more likely inherent to the language, and not the phenomenon?

Yes, I would agree although we might discuss what is meant by ‘properly explain’.

I would also argue that mathematics can most certainly describe the numinous and the transcendent; it simply is that the descriptions seem to cheapen the personal experience or seem to be inadequate so one is opt to dismiss as not applicable; when in fact electro-chemical reactions in one's brain is about as mathematical as it can get

I should be grateful if you could develop each of the three parts of this statement further.

Physics needs mathematics. It is the only unambiguous language that we have been able to invent, the one that is most logical.

While I agree with the second sentence, it is the implications of the ‘need’ part that puzzles me. I will develop this in my reply to Pythagorean.

Euan

10. Jul 30, 2008

### Legion81

I can say something about 2.

At the fundamental level, math is the only "real" way to work with the universe. As Descarte's wax theorem stated, everything, even senses, can deceive you into thinking something is real when it is not. But math is real because it is a concept... based on zero and one. All numbers are a combination of one, so you can have a pure thing to work with. When we explain our universe, the laws of math give us a "real" answer that cannot be incorrect, unless of course, if the person working with the math does not do it right...

11. Jul 30, 2008

### will.c

Math can only give answers that are consistent. The "truth" of mathematical statements is internal to the system being used. The universe (external to the system) doesn't care about our mathematics.

12. Jul 31, 2008

### Coldcall

Well this is an interesting subject. I just read Marcus Chown's last book "Never ending days of being dead" - great title. And one chapter is spent on the idea that a computer program would be far more capable of explaining everything in the universe than maths.

In this chapter Wolfram (Mathematica founder) argues that maths is actually very limited in only being able to calculate certain types of problems and most of those have already been tackled. So his view is maths is limited to what he claims are easy problems.

13. Jul 31, 2008

### Euan

Pythagorean, Legion81, Will.c (and with further reference to Robertm)

One of the implications behind all replies so far is that there is a difference, a separation, between the phenomena and their description; in short, they are not the same thing.

Stripping it down to bare essentials, we have two separate streams of activity: observing (elements/aspects of) the universe (=physics through experiment) and an internal (completely logical and consistent) process of reasoning (=mathematics) that a priori has nothing to do with the observed universe. The second is then used to ‘model’ (Pythagorean) or ‘explain’ (Robertm) the first and this is the only sure way to do that because other methods may ‘deceive’ (Legion81).

Furthermore, the second has the advantage (over other descriptive measures, presumably) that it allows ‘quantitative predictions’ (Borek).

I hope this is a fair summary so far.

While, of course, I appreciate that mathematics is spectacularly successful in explaining/modelling/predicting phenomena in physics, I still do not see why this should be so. After all, there are many other separate streams of activity where the one does not explain/model/predict the other. So, at the deepest level, why are physical phenomena – which we are assuming here are real and can be observed – mathematical if, indeed, at the deepest level they are so?

Take Pythagorean’s Inverse Square Law example. If the phenomena that this relationship models is (in some way as yet not clear to me) synonymous with the phenomena themselves, then (I would assume that) any and all defined and universally accepted operations I conduct on the mathematical relationship will result in further relationships each of which precisely models some other distinct behaviour or aspect of those self-same phenomena.

Is this true? Is it what happens? And, if not, what then at the deepest level is the relationship between mathematics and physics?

Two (among many?) simple possibilities are: (i) there is a common parent of these two streams of activity (they both take place in the brain, for example), or (ii) in some very deep sense, physics at its most fundamental level is indeed completely synonymous with mathematics. I suspect the second is the case but (to date) have not found the ‘language’ that ‘proves’ this.

Euan

PS: Coldcall - thank you for your response. I had prepared this posting before I read it and so have not as yet responded to what you say.

14. Jul 31, 2008

### octelcogopod

Euan,

A language is only a model that has symbols that represent something in the world we see.
When I say in english, "that guy just jumped off a plane with a parachute" I have successfully explained a physical phenomena.
Similarily, with math, if I say there are /two/ apples on the table, I have used a very basic quantitative statement to describe the world.
I think the problem you have, is that when these mathematical equations become complex, you lose a bit of perspective on what is really happening.

No matter how complex the language gets, they are still only symbols from which the brain has created based on what it observes.
Physics and math alike, are both models.. Models that occur in the processing in a human brain.

Mathematics is not synonymous with physics, because math is an abstract language that can be applied to anything quantitative, and physics is more specified to the perceived world.
In some ways everything we have is the perceived world of course, but that doesn't necessarily make it physics.
I would say math is a bit broader than physics.

Furthermore, when we apply physics, we are actually using it in the real world.
We are not only doing math, we are also creating a model for something physical.
Physical reality is by default a quantitative place. Everything we know so far, can be described with "what did it look like and how many were there, how big was it" and so forth.
math is the side effect of this brain processing, where (mysteriously enough) 1 + 1 = 2 always.. This is just how reality is..

15. Jul 31, 2008

### robertm

Yes, I am afraid that a discussion as such could turn into to quite a spiel.

Certainly,

Emotional responses experienced by a consciousness are grounded in reality, i.e. they are purely physiological. So far, we have been able to quantify (describe in other words) any aspect of reality using the language of mathematics.

Experiences of the numinous are simply complex physiological interactions grounded in reality, hence, could be fully quantified using mathematics.

Many factors including the human ego, a general lack of understanding, and the fact that the human brain is poorly wired for mathematics; provides a common feeling of inadequacy when attempting to quantify ones own personal subjective experiences. I think it mostly can be attributed to the ego. "We must be the chosen people." "The earth is the centre of the universe." etc...

Good summary octelcogopod,

I would say Euan, that the apparent relationship is a mixture of your two points. Physics and mathematics are certainly nothing more than human invented concepts; however, (as octelcogopod suggested) the universe (afa we know) operates in a logical sequential mathematical manner whether humans are there to quantify phenomenon or not. We can be most certain that the tree does indeed make a sound when falling in the woods, no matter who or what is there to 'hear' it.

16. Jul 31, 2008

### Eidos

This is a great thread, thanks to all of you :)

As an aside:
Modeling is an approximation to what actually happens and has boundaries where the model no longer applies. Eg: Newton's Laws apply well at low velocities, whereas for velocities that are fractions of the speed of light, Einstein's relativity would need to be used since Newton's formulation predicts incorrect results i.e. are not physically observed.

We often extrapolate from some basic premises and 'follow the maths' to see where it leads us. This would constitute (very roughly speaking) a theory, which would contain a number of predictions.
An example given earlier could clarify what I mean by this:

$$f(r)=\frac{1}{r^2}$$

is a model for the force acting on an object by gravity, at a distance 'r' away. This function predicts what the force will be for all 'r'. The way to verify its predictions is to take a series of measurements.

So
$$f(100)=\frac{1}{100^2}=0.0001\, N$$

We then move 100m away from the earth and take a measurement of the force. If these two numbers agree then we are in good stead. We do this a (preferably large) number of times and if the numbers continue to agree, then we have successfully modeled gravity, over the range of distances measured, using equation f(r).

Last edited: Jul 31, 2008
17. Aug 4, 2008

### Euan

My thanks to everyone who responded; you have given me a good deal to read and think about.

As I said at the start, I am neither a physicist nor mathematician; you are all far more expert than I am. But I still do have a nagging feeling that there may be a different way at a fundamental level of looking at the relationship between these two disciplines, one that throws a different light on why mathematics is so spectacularly successful in physics. Perhaps I could return to the topic at a later date once I have followed up all the leads you have so kindly provided.

Thank you all

Euan

PS to Robertm: thank you for your second posting where you develop your 'emotional response' argument. This, too, is something I should like to pursue more at a later date.

18. Aug 6, 2008

### BWV

I have this suspicion that we find this question interesting because the math is challenging. To a sufficiently advanced (i.e. non-human) intellect perhaps the math behind physics is a set of simple tautologies and the question of why there is a link between physics and higher math is no more profound or in need of explanation than asking why there is a link between addition and accounting.

We do not think it profound that addition predicts the amount of change in our pockets, perhaps the relationship between gauge theory and quantum mechanics it is equally mundane

19. Aug 7, 2008

### Fredrik

Staff Emeritus
I'm surprised that no one mentioned set theory as an answer to this. The ZFC axioms of set theory can be used to construct the integers. The integers can be used to construct rational numbers. The rational numbers can be used to construct real numbers. The real numbers can be used to construct complex numbers. And so on.

One of the reasons is that we need mathematics to properly define the concepts we're talking about. Concepts such as "particle" are hard to define in terms of other things in the real world, but we can give them exact definitions within the framework of a mathematical model. Exact definitions are of course necessary to make sure that different physicists who are using the same word are talking about the same thing.

All I can say is that I don't think it's surprising, considering that the main ingredient in every theory of physics is a mathematical model.