How do mathematical proofs prove anything in the physical world?

Factual Questions
1

The title says it. I don’t understand how numbers on a whiteboard put through ridiculously complicated convolutions can be used to prove a physical concept. It just doesn’t make sense to me, as mathematics are a human construct and thus subject to the limits of existing as an abstract concept. Even basic arithmetic doesn’t make much sense in relation to real objects. “2” for example also often means “twice as much” except that it doesn’t unless the objects are exact physical clones of each other. So to my mind, using “2” to represent quantities of anything beyond dollars,(also an abstract concept), just doesn’t seem right. Further, how does making the numbers work out justify the probability of an untested physical concept?l

Since Math is used to do so much theoretical research i’d really like to understand it better, but my dyscalculia along with a bitter distaste for maths has prevented me from gaining a better understanding of these fields. I want to fight my ignorance here! Can anyone walk me through this stuff gently?

Symptoms I exhibit if it’s relevant to your explanations:
* Difficulty with times-tables, mental arithmetic, etc.
* May do fairly well in subjects such as science and geometry, which require logic rather than formulae, until a higher level requiring calculations is obtained.
* Problems differentiating between left and right.

  • An inability to read a sequence of numbers, or transposing them when repeated such turning 56 into 65.
    • Difficulty with games such as poker with more flexible rules for scoring.
    • The condition may lead in extreme cases to a phobia of mathematics and mathematical devices.
2

My thoughts are somewhat similar, Acid Lamp. Surely taking into account the versatility of the worlds spoken languages, it shouldn’t be impossible to explain any mathematical equation, or concept etc, into words a reasonably intelligent person can understand?

3

The answer is that mathematical proofs prove nothing about the real world.

However, mathematics can be used to model the real world, i.e., to construct theories about how the world works and make predictions about what will happen. But, even if those predictions are right, you still have proven nothing: you just have a theory that apparently works.

4

Now here is where I sound Profoundly Ignorant: How?

How do we make abstract numbers accurately model something as complex as a physical object? How that object interacts with it’s environment and so forth…

5

Generally, you are not modelling every aspect of the object, only the part relevant to your theory. So, in an economic model, you might have prices of goods and the quantities of those goods sold to consumers, and ignore any other properties of those goods or those consumers, e.g., the colour of the goods, or the religion of the consumers. You simplify, to understand just one or two aspects of reality.

6

This is known as the unreasonable effectiveness of mathematics, and is the major open problem in the philosophy of mathematics. So don’t feel too bad for not getting it.

7

A lot of mathematical concepts can be explained to a non-specialist, but I’m not sure that anyone outside of a mathematical discipline can appreciate them. For instance, I’m sure that I can explain what a module is reasonably well, but I don’t think I could ever tell you why pure mathematicians care about them.

8

Of course, and this is possible. However, it’s unwieldy working with huge sentences when a simple shorthand will do.

9

It’s a matter of translation - the mathematical language is much more formal and rigorous (and concise) than any spoken language could ever reasonably be. It is possible for a well-honed analogy to illustrate some deeper principle though - I remember a lecturer introducing us to special relativity with the concept of a photon bouncing between two perfect mirrors on a passing train.

10

I just clicked on ultrafilter’s module link. The first paragraph of the wiki article appears to be written in English, but I don’t understand a word of it.

Vector space?!! Scalars?!? THE?!!???!

11

Acid Lamp, your question is imprecise, sufficiently so to make answering it difficult.

What do you mean by “mathematical proof?”

All mathematics relies upon certain basic assumptions that are not provable, but instead act as a starting point from which other things can be proved. Thus, for example, in Geometry, certain basic assumptions are made, from which others follow. When the assumptions turn out to be inapplicable to the actual world, the applicability of the conclusions becomes similarly limited. Thus, for example, the inability to accurately run a GPS sattelite system on the basis of Euclidean models.

The key is to make assumptions that are valuable in describing whatever it is you are attempting to model with math. For most of us, that modeling doesn’t have to get much past rational numbers (numbers which are the quotient of two integers, the divisor not being equal to 0). Irrational numbers (numbers whose digital representations do not “end” or “repeat”) are enough to make one break out in sweat if one thinks about them too hard (show me π of something, please!). And complex numbers simply boggle the mind!! But we can do math with them, and we get results that are helpful to understanding the real world, and predicting what will happen when we build certain things, or take certain actions. Nevertheless, the fact remains: if we don’t manage a correct modelling, then garbage in results in garbage out. See, for example, the de Havilland Comet. :wink:

12

You use a LOT of numbers. But most of the time (in fact, never) do we actually need to fully model every characteristic of an object.

Let’s take an example. We want to model how far an object will fly if thrown from a catapult. Well, we need to know a few things. We need to know the vertical force of the catapult toss, and the horizontal. We need to know how strong gravity is. We will need to know the object’s weight (not its mass). We can probably disregard air resistance, although knowing that would help us get a better model.

Now, through experience and testing of other people (done over centuries), wehave some forumlas which tell us how to use these numbers. or we could work it out ourselves. We know that vertical and horizontal energies are unrelated and ignore each other. So, if we figure out how far up the object will go, we now how long it will be until it hits the ground. Sinc we know how long that is, we can take into account the vertical motion over that period and determine how far the catapult will throw the object.

The abstract numbers really aren’t that abstract. We have a default (and arbitrary) measuring system, but that measuring system is irrelevant so long as its accurate.

Modeling is all about finding a formula which expresses how things in the real world behave.

Another, simpler example: A ball rolls down a track. You take a stopwatch and measure how long it takes to do so. Now, keeping the track at the same incline (that is, how it slopes down), we extend its size. It’s now double the previous size. How long will the ball take to roll? You would say double the time. You would probably be pretty close.

13

Your link seems to be discussing rings, fields, and two sorts of math, one apparently describes how people get to work, the other has to do with gender studies. I didn’t know that algebra related to the gay farmer marriage debate as well as the fuel crisis. :wink:

In all seriousness though, I couldn’t follow a bit of that wiki, or any of it’s linking topics which all seem to require that I already comprehend whatever language it is that mathematicians speak.

14

You might ask the same question about how a physical thing can be represented using any language.

How are instruction manuals for ovens possible? How is something as infinitely complex as the physical oven captured with any accuracy by a few squiggles on paper? The manual says that I can open the door of the oven by pulling on the handle, and, lo-and-behold, I can. Somehow, the physical concrete properties of the oven are represented by artificial abstractions, texts and diagrams, in the manual.

In my view, mathematics is just a special case of this. Mathematics, at least when it’s applied to the physical world, is just that fragment of our language that we use to talk about, e.g., magnitudes of things instead of whether they are oven-door-handles or oven-doors.

You wrote that it’s inaccurate to say that there are two X’s unless the X’s are perfectly identical. I don’t see a problem here, any more that there is a problem with calling two distinct things “oven doors” even when they aren’t perfectly identical to each other. That is, it is no more a mystery than the fact that the same oven manual can be used to understand the behavior of several distinct physical ovens.

15

I mean how does a “eureka” moment with numbers on a board translate so neatly into a new type of energy or a particle type too small to be observed?

16

It doesn’t. Those have to be verified empirically. You never hear about all the sums that didn’t work out.

17

Well let’s go back to apples.

counting them up is useful only of you need to know the number of apples you have. If you want to interact with your apples chemically or with physics, wouldn’t one need a more precise definition than “2 apples”? I thought maths were supposed to be rigid with little room for error, and that what was made them so accurate in modeling. You seem to imply that we play fast and loose with them, and for most things that’s “good enough”.

18

Gotcha. What I’m asking I suppose is how one can use those numbers to accurately predict the result of, or even conceive of the idea of that empirical experiment ?

Right now my understanding of theoretical science goes like this:

  1. Notice a slight aberration within something physical
  2. ?
  3. loads of maths
    4.?
  4. experiment to prove your theory
  5. profit!
19

Let’s see if I can take a shot at this. Forgive me if I’m repeating any previous posters.

Proofs I’ve done in the past don’t mean that I’ve proved that my formula relates to the world, they prove that one side of the equation is equal to the other side. This is useful when we want to “massage” our formula to get it into a form we can use. In the real world, some values are going to be harder to measure than others, and some easier, in different situations. We need to massage our formula to be useful if we can measure, say, pressure and temperature, but not volume; or if we can measure volume and temperature, but not pressure (PV = nRT, ideal gas law). In other words, we’re not proving that our equation is correct, only that it works out based on the established rules of mathematics.

When you say numbers are pretty arbitrary, you’re exactly right. We assign numbers to things based on how it will be useful for us. For example, 0[sup]o[/sup]C is the freezing point of water, and 100[sup]o[/sup]C is the boiling point. On the other hand, 0[sup]o[/sup]F was just the coldest it got in Poland (Germany?) one winter, and 100[sup]o[/sup]F is the temperature of the human body (Dr. F was a bit off). Now, these numbers work great for you and I telling each other how hot it was yesterday, but we can’t use them in calculations because 20[sup]o[/sup] isn’t twice as hot as 10[sup]o[/sup]. Instead, we have to go to Kelvin or degrees Rankine, which both have absolute 0 as their starting point. Therefore, 100 Kelvin -does- have twice as much energy as 50 Kelvin. In other words, they’re pretty arbitrary.

To move on to the actual math part, in order to “prove” an equation, we have to do lots and lots of testing. There are some fields, such as fluid mechanics, where the phenomenon still aren’t entirely understood, so instead of being able to derive the equations theoretically, we have to to the testing to obtain empirical data, which we can then use to form an equation around. We can’t really use the numbers to justify an untested concept without lots of observation.

Finally, when we talk about calculus, derivative simply means “rate of change”. So, the rate of change of your position is your velocity*, while your rate of change is your acceleration. Therefore, your acceleration is your second derivative of your position.

*Velocity is a vector, that is, a magnitude (speed) and a direction (north), while speed is a scalar that only gives the magnitude.

20

Here’s a simpler example. Everyone knows how to calculate the area and volume of 2 and 3 dimensional objects. That has direct relevance to the real world. But this can be extended quite easily to objects of any number of dimensions. Trying to think about this without the mat makes my head hurt - with just the math it isn’t too bad. is this actually relevant to the real world? Well, string theory is talking about 11 and 12 dimensional space, so maybe.

Another example. Clearly i, the square root of -1, is bogus. It’s even called imaginary. But it actually gets used all the time in describing waves. Pretty amazing, once you think of it.