How do mathematical proofs prove anything in the physical world?

Factual Questions
21

So if I’m understanding this correctly,

according to Santo Rugger: We are sort of making it up as we go along, but it is good enough to be getting on with. No one really knows WHY numbers are so accurate either as posted by Ultrafilter, but I shouldn’t worry too much about it since hardcore math people don’t know either; and the system works well anyway.

Cool I can accept that.

Now I still need to get those blanks filled in the process. How do those numbers become empirical experiments, and how do the results of those experiment become benefits to humanity, other than the tacit benefit of increased knowledge that is.

22

12 dimensional space is so cool. :smiley:

It is beyond amazing, it makes no sense. If the number is clearly a fudged answer then how can we accurately build upon a false foundation. Or is it that we do not yet understand the nature of i ?

23

Mathematics is the language of physics (and by extension, all natural sciences on a fundamental level). Of course, like any language, you can assemble the words in ways that bear no relationship to anything real or intelligible, even though they are constructed in a correct manner (see Lewis Carroll’s The Hunting of the Snark for an example of this in English). The way you use math to describe anything in the real world is that you “model” a system–that is to say, you create a series of equations based upon whatever parameters and interactions you assume to be relevant–and then you use this model to predict the result of known interactions. If the predictions are true up to the limits of measurement (and repeatable) then the theory is sound, at least, until someone comes along with a more comprehensive theory that works. The mathematics itself, however, isn’t “real”; it’s just a grammar that describes fundamental relationships var more succinctly and explicitly than any natural language can.

Can you convert a symphonic score into English? And if you did, would musicians be able to reproduce music from it? How would you describe, in explicit, useful mechanical detail, how a butterfly’s wings move? And our description of things that are well-described mathematically but differ so much from everyday experience that we can’t even properly conceive of them–such as the quantitized behavior of fundamental particles or the relationship between space and time described by Special and especially General Relativity–are only very rough analogies, approximations so coarse that even trying to render them in English results in seeming paradoxes.

Math, like music, is a language designed to explicitly discuss fundamental mechanics. One would not attempt to write a sonnet in it, and jargon in natural language which attempts to bridge between math and natural language ends up being very clunky in order to have the same degree of uniqueness and explicitness as math (“If and only if,” “The dependent conjunction of two homogenized toplogies,” et cetera).

As for understanding math, it is true that some people are more naturally adept at it than others, just as some people are really good with music, or poetry, or whatever. You just have to start by accepting the premises and definitions, just as you learn German by conjugating verbs and learning genders before being able to form complex sentences.

Stranger

24

Think of it this way. Math involves the manipulation of symbols according to certain rules. Programs have been developed to prove theorems, for example. Now, you might set up a computer to apply all sorts of transformations on known theorems and hope to come up with something, but that would be like the monkeys trying to type Shakespeare. What actually happens is that someone has an insight, and then tries to write a proof to see if the insight is true. This is usually done step by step, so you try to prove something simpler first. Great mathematicians have great insight and a great ability to be able to manipulate the symbols. And a good understanding of other people’s results and how they can be applied. Ferynman wrote that he just saw calculus, at an early age. I sure don’t.

You might be interested in reading the biography of Feynman - I think it is called Genius. He had the question of how light knew to travel in a straight line. IIRC, there were mathematical models of this, but they all resulted in lots of infinities all over the place, and clearly weren’t right. He came up with the way to describe the problem and get the answer - and I better leave it there for someone who actually understands it.

25

If you don’t believe it, try to find a translation of Euclid or Archimedes that uses verbal explanations rather than modern mathematical notation, the former being what the original authors used since they didn’t have algebraic notation.

“If three quantities be taken such that the proportion of the greatest to the medium is as the medium to the least…” (or a/b=b/c to us).

26

The only thing I know about Feynman is that he traveled to Tuva. I’ll look into the book. :slight_smile:

27

Essentially he assumed that you could have terms in only certain increments, and that most of these would end up canceling each other out (renormalization), leaving only a handful of components with divergent terms that you had to sum up to obtain the resultant direction. The math is really, really complicated–even for people who do math for a living–but the concept is actually quite simple. Feynman himself describes it reasonably well in *Q.E.D.: The Strange Theory of Light and Matter* without resorting to any but the most simple equations (albeit not in the same manner as presented in physics texts).

As for imaginary numbers, it is somewhat unfortunate that the term “imaginary” was used (due to being created by pure mathematicians before any real use had been determined for it); while we often talk about time-varying properties “phasing” through the real and imaginary plane, instead of using Re and Im to lable the axes we could have used X and Y, or A and B, or Up and Right, or any labels we choose. There is nothing fakey about imaginary numbers except for the name. In quaterions, which are essentially the concept of imaginary numbers extended into a 4-space, we just use {1,i,j,k} for directions and nobody worries that the letters stand for something.

Stranger

28

Although math may not be able to prove physical facts, it is very good at modeling them, has has been mentioned. Sometimes the mathematical implications are quite surprising, like the fact that the numbers of opposing spirals in composite flowers, pine cones, and the like, are adjacent numbers in the Fibonacci sequence. In turn that sequence is intimately related to the Golden Proportion, the number whose square is one more than itself.

29

Acid Lamp: Now I still need to get those blanks filled in the process. How do those numbers become empirical experiments, and how do the results of those experiment become benefits to humanity, other than the tacit benefit of increased knowledge that is.

“Filling the blanks” is the process of mathematical modeling and it’s a difficult activity to explain to someone without a bit of mathematical training. A model is a mathematical representation of a problem or situation, oftentimes involving simplifying assumptions to make the mathematics tractable. This is as much an art as it is a science. And, yes, there’s some trial and error involved, but experience will lead to fewer errors until a usable model is derived. And then it needs to be tested with real experiments. If the experimental results are consistent with the model, then the model lives on. If not, the model requires revision. After a while if enough empirical data supports the model, then the model can be used to predict outcomes. And if further experiments verify these predictions, then the model will gain even more credibility. When a mathematical model is “proven” to a high enough degree, then it may be elevated to the status of a “theory” and maybe even eventually to the exalted status as a “law”. Surely, you’ve heard of Newton’s Law of Gravitation and you must admit that gravity has had quite an influence on your life!

30

I’m not sure where you’re getting the “fast and loose” from. I’m not saying that math is any more “fast and loose” than, say, the oven manual is about which part of the oven is the door.

I’m having a hard time pinning down exactly which kinds of physical explanations you find mysterious. Which of the following observations best captures the mysteriousness that you are trying to describe?

(1) A theorem of arithmetic states that 5 + 7 = 12. And, amazingly, when I have five apples, and I add seven apples to them, I find myself with exactly twelve apples.

(2) The Pythagorean theorem in Euclidean geometry states that, in a right triangle with legs of length a and b, the length of the hypotenuse is the square root of a[sup]2[/sup] + b[sup]2[/sup]. And, amazingly, when I have a real physical right triangle (as established with a plumb line and level, say), I find that the lengths of its edges (measured with a ruler) approximately satisfy the Pythagorean theorem.

(3) Consider a stone flung into the air. Using Newtonian mechanics, you can write down a mathematical expression P(t), which, if you evaluate the expression for a particular value of t, will yield the position of the stone at time t.

(4) Consider the stone and the mathematical expression from (3). It turns out that there are certain purely mathematical operations that we can perform on the expression P(t) (e.g., computing its derivative) that give us information about purely physical properties of the stone (e.g., its velocity).

(5) Observations like (1) – (4) aren’t just true, but, moreover, we are somehow able to know that they are true with sufficient certainty to make the enterprise of science worth the trouble.

31

Here’s a link to the most famous article on this subject, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”:

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

32

I just want to say that mathematical models always involve simplifying assumptions. That is the first rule of applying math to the real world: Throw out all the information that you don’t need and measure/calculate/manipulate the rest.

That seems to be Acid Lamp’s biggest issue to me.

That’s pretty much it. It seems obvious that if you need to know anything else about the apples, you have to find that out some other way besides counting them. Like weighing them, or measuring the wavelength of the light reflected off of them, or squeezing them, or x-raying them, or dissecting them or whatever. ETA: You of course shouldn’t do any of this stuff if you don’t have to. You don’t need to break out the spectrum analyzer to find out how many apples you have or if they taste sweet.

People need to understand that math is about simplifying things, not making them more complicated.

Also, i is just as real as any other number out there. Its name is the only imaginary thing about it.

33

Adding to that, perhaps it adds clarity (or perhaps not, but I’ll give it a shot anyway) to think in classes and instances: each individual apple is an instance of the class ‘apple’, disregarding its other properties (colour, taste, texture etc.). For the problem count the number of apples, only the information of whether or not an object belongs to (is an instance of) the class ‘apple’ is necessary, so we can disregard any other information without any losses in respect to the given problem.

That’s essentially a simple form of mathematical modelling of a physical problem, and using this model, we can derive (true) statements about physical reality: if we remove 3 apples from our stash of 7, we’re left with 4.

Now why, you might ask, does this operation we’ve just performed – subtraction – relate to the physical world at all? Basically, this is due to the construction of the natural numbers, also called counting numbers: each natural number has a successor which is also a natural number, and the natural numbers are well ordered, which means that for two numbers m != n (!= : ‘not equal’), either the relation m > n or m < n must hold (so, one of them’s the bigger one).

This means we can count the apples: we identify the smallest number of apples we can possess with the smallest element of the natural numbers (0 or 1, depending on whether you’d say that no apples is a number of apples as well – let’s just go with 1, the reasoning will be identical in this case), and gradually add apples to our stash, and identify each subsequently larger number of apples with the next element in the natural numbers – 2 apples, 3 apples, and so forth.

Then, we can define the addition of apples: adding a given number of apples to our stash is the same as adding the two representative natural numbers (the amounts of apples in each stash), because if you add 4 apples to 3 apples, it’s the same as adding one apple 4 times.

And subtraction, then, works exactly the same way, only by removing apples, and using the predecessor in the natural numbers as representation of your amount of apples until you’ve got none left.

Now, why did I go through all of this in all its ridiculous detail? Because, essentially, that’s how all mathematical modelling works: identifying a physical property (number of apples) with a mathematical object (natural number), and then manipulating that object according to the rules of an appropriate structure (the set of the natural numbers, in this case).

Thus, if your identification is correct, you can derive valid predictions from your manipulation of abstract mathematical concepts, the same way we can say that we’ll have 4 apples left from our previous 7 if we eat 3.

34

Nitpick: In technical terms, you appear to be discussing the property of being “linearly ordered”, also known as “totally ordered”. That the natural numbers are furthermore “well ordered” is the stronger claim that, to put it one way, there are no infinitely descending sequences of natural numbers.

35

You’re right, of course, I tend to fudge my jargon a bit sometimes.
But IIRC, the way I phrased it merely means that I neglected mentioning that every non-empty subset of N has a least element, since a well-ordering is a well-founded total ordering. Doesn’t it? Umm… I meant well? English isn’t my first language? :wink:

36

Yes, that’s right. I tried to explain what well-foundedness was in more intuitive terms by using “no infinitely descending chains”, but it’s all the same. [Except when it isn’t; my definition presupposed a little bit of Choice. Even the “…has a least element” definition of well-foundedness doesn’t quite directly address the salient properties (and fails to generalize properly outside of classical math). Really, the best definition of “well-founded” is “The sort of thing that you can do induction over”, suitably formalized.]

37

I can’t say that I “get” the hoo-haa over the “unreasonable effectiveness of mathematics in the sciences” (there’s also a related paper “the unreasonable effectiveness of logic in computer science”)—but that’s not surprising, a lot of problems in philosophy seem to be molehills, rather than mountains, to my untrained eye.

Why would it surprise anybody that mathematics is extremely good at describing physical phenomena? Most of our mathematics comes from observing the world and abstracting from it, or from observing patterns within mathematics itself, and abstracting them (c.f. abstract algebra). It isn’t a coincidence that 1+1=2 in Peano arithmetic, just as one apple grouped with another gives a group of two apples.

38

P’raps a slightly different example, from early geometry: the point and the straight line. In mathematical theory, a point has no width or length. A straight line is perfectly straight and has length but no width. Right? This model has been extremely useful in devising things like measurement, architecture, etc and a host of other practical applications.

However, in the real world, there is no such thing as a “straight line.” You can’t draw something that has no width. You might be able to draw something that was only an electron-wide, but that’s still width, however small. So, the mathematic model is extremely useful and provides many practical applications, even though it does not match the real world.

Similarly, the Pythagorean theorem (and related Trigonomety) allows for many practical applications, such as calculating distances/heights that you can’t actually measure. (If you know the length of the shadow and the angle between the tip of the shadow and the top of the tree, you can calculate the height of the tree.) Or the formula to determine the circumference of a circle if you know the radius. However, such calculations often involve “irrational numbers” like the square root of 2 – an infinite, non-repeating decimal. Such numbers exist in mathematical theory, but not in the real world. There is no way to draw a line exactly 3.1415926535897932384626433… inches long. Measurement tools and drawing tools just aren’t that precise. Again, these are things in the mathematical model that are extremely useful and have many, many practical applications… even though they only exist in the mind, in the theoretical realm.

That help?

39

C K, was that addressed to me?

40

I think you have it backwards, but I’m a little out of my element. Wasn’t there a theorem that states that the probability of choosing a rational number at random on the real number line is equal to pretty much zero?

I’d say our measurement tools simply aren’t precise enough to find actual rational numbers in the real world and that every number we measure is irrational.