How do mathematical proofs prove anything in the physical world?

[Acid Lamp]

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 (http://en.wikipedia.org/wiki/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.

[ivan astikov]

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?

[Giles]

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.

[Acid Lamp]

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.

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...

[Giles]

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...
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.

[ultrafilter]

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...

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.

[ultrafilter]

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?

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 (http://en.wikipedia.org/wiki/Module_(mathematics)) is reasonably well, but I don't think I could ever tell you why pure mathematicians care about them.

[Capt. Ridley's Shooting Party]

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?


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

[Baron Greenback]

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?

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.

[Flander]

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?!!???!

[DSYoungEsq]

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. ;)

[smiling bandit]

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...

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.

[Acid Lamp]

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.

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. ;)

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.

[Tyrrell McAllister]

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...

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.

[Acid Lamp]

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

What do you mean by "mathematical proof?"




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?

[Capt. Ridley's Shooting Party]

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

[Acid Lamp]

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.

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".

[Acid Lamp]

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

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.?
5. experiment to prove your theory
6. profit!

[Santo Rugger]

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, 0oC is the freezing point of water, and 100oC is the boiling point. On the other hand, 0oF was just the coldest it got in Poland (Germany?) one winter, and 100oF 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 20o isn't twice as hot as 10o. 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.

[Voyager]

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.
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.

[Acid Lamp]

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.

[Acid Lamp]

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.

12 dimensional space is so cool. :D

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 ?

[Stranger On A Train]

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.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.

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?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

[Voyager]

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.

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.

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.
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.

[Spectre of Pithecanthropus]

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

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).

[Acid Lamp]

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.

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

[Stranger On A Train]

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.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 (http://www.amazon.com/QED-Strange-Princeton-Science-Library/dp/0691125759) 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

[Spectre of Pithecanthropus]

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.

[nivlac]

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!

[Tyrrell McAllister]

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.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".

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 a2 + b2. 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.

[Wendell Wagner]

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

[DrCube]

Acid Lamp: A model is a mathematical representation of a problem or situation, oftentimes involving simplifying assumptions ...
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.
Well let's go back to apples.

counting them up is useful only of you need to know the number of apples you have. 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.

[Half Man Half Wit]

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.
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.

[Indistinguishable]

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).
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.

[Half Man Half Wit]

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.
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? ;)

[Indistinguishable]

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.
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.]

[Capt. Ridley's Shooting Party]

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.

[C K Dexter Haven]

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?

[Capt. Ridley's Shooting Party]

C K, was that addressed to me?

[DrCube]

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.
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.

[Acid Lamp]

This has all been fascinating, helpful, and frustrating to me all at once. What a great thread!

[ultrafilter]

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.

Most of the unreasonableness has to do with quantum mechanics. For instance, we know that energy and mass are related by one equation, and energy and wavelength by another. So you mash them together and you get an equation relating mass and wavelength. This looks like nonsense, but it turns out that particles actually do exhibit the mass-wavelength relation that was derived.

[DrCube]

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.
The only thing I could find to back myself up on this was Wikipedia's article on rational numbers (http://en.wikipedia.org/wiki/Rational_number):

The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set.

I take this to mean that if I throw a dart at the real number line, I am going to hit an irrational number with 100% probability, even though there are an infinite number of rationals there. There are just so many more irrationals.

I just like to bring stuff like this up occasionally, especially when people say things like "Clearly i, the square root of -1, is bogus." In many ways, real numbers are far more difficult and counterintuitive than complex numbers.

I like Isaac Asimov's reply to his philosophy professor, when asked to produce i pieces of chalk: "Well, can you show me 1/2 a piece of chalk?" It just ain't possible.

I realize on preview that this whole post has been a hijack, and probably makes the OP's head spin, but I feel it is interesting and I'm posting it anyway. :p

[Stranger On A Train]

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. "We assumed the horse to be a sphere to make the math easier."

Stranger

[Indistinguishable]

I take this to mean that if I throw a dart at the real number line, I am going to hit an irrational number with 100% probability, even though there are an infinite number of rationals there. There are just so many more irrationals.
Yes, yes, but there's also a 100% probability that you don't hit π, to back up C K's point. But then, there's also a 100% probability that you don't hit 5, (to undermine it?).

The idea that the real world contains rationals but doesn't contain any irrationals is really odd. [The idea that the real world contains numbers at all is weird speech, it seems to me, but that's a different issue]

[Hypnagogic Jerk]

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?
Well yes, given that the Lebesgue measure of the rational numbers is 0, if you have a uniform distribution on the reals, the probability of choosing a number at random and getting a rational is 0. (Doesn't mean that it's "impossible", though; that's the thing with uncountable probability spaces.)

But once again, we're in an idealized mathematical world here. The real number line doesn't exist in the "real world". I think what Dex meant is that we live in a world that's really in fact discrete. We could make the argument that only the natural numbers exist in the physical world.

Of course, that depends on whether we can approximate the physical world with mathematics, or rather the physical world is an approximation of the idealized mathematical world. ;) For example, the OP said that mathematics is a human construct, and I get his point, but there is a philosophical debate whether mathematics is really "invented" or "discovered". Some mathematics seems so natural that we can't but wonder whether it exists independently from us. Would an intelligent alien species have the same math as us? I think we've discussed this question before on this board.

[Voyager]

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
I wouldn't use fakey as a synonym for imaginary. I think i is a great example of the kind of thing the OP was asking about. It is perfectly reasonable to ask what the square root of -1 is, and to assign a symbol for it. It is also perfectly reasonable to manipulate it. The interesting thing is that i, which seems to be purely symbolic and which isn't something you can point to directly in the real world, actually is important in describing real world things. It was the simplest example I could think of for something that seemed to be just mathematicians playing but which is at the heart of reality.

[Indistinguishable]

Of course, the very fact that it is capable of being used to describe real world things means we can point to it in the real world, just as much (and as little) as we can point to any other mathematical object in the real world.

[Bear_Nenno]

This has all been fascinating, helpful, and frustrating to me all at once. What a great thread!
I think people are being way too complicated here.


1. Notice a slight aberration within something physical
Notice how an object that is released from a particular height will fall to the ground.
2. ?
Start your critical thinking and consider possibilities. First, lets name this force... we will call it gravity. Now let us try to figure out what we want to learn about this new force called gravity. How powerful is gravity. How fast does it make objects fall? Does it cause them to fall at a constant speed? Or does it make them accelerate (ie, does the object keep speeding up the longer it is falling?)

Now perform some experiments dropping various objects from various heights. Measure the distances and record the time it takes to hit the ground.

3. loads of maths
Start calculating all the data you collected. Your data tells you that objects--no matter how heavy, all fall at the same speed.... interesting, you say.
Your data tells you that objects do not fall at a constant speed. They seem to accelerate... that is, they reach faster speeds if they fall from higher up. You only measured falls from a height of 1000ft, so you haven't learned about terminal velocity or wind resistance.

4.?
Now you can use your knowledge and your math to make predictions on the REAL WORLD!!

For instance. Even though you never dropped a 500lbs anvil from 2000ft, you can use your math skills and your previous data to calculate how fast the anvil will go, and how long it will take to hit the ground.

5. experiment to prove your theory
You work the math, and come up with your numbers. Now go ahead and drop it from 2000ft. Voila. Your math was correct and you used it to predict something in the real world.

6. profit!
But wait.... one of your colleagues just got back from the moon. All of his data is different than yours. Things do not fall as fast on the moon he tells you.
So you guys start to think about this and realize (through math) that the force of gravity on the Earth is 6 times as strong as that on the moon.
After measuring the moon, you realize that the Earth is 6 times more massive than the moon. So you think this is the cause.

So now you can make a hypothesis that the massof the planet effects the gravity. You can now make mathmatical predictions about the strength of gravity on... say Mars. Or Jupiter.


This is it in the simple form. Even though we haven't been to, say, a black hole. We can use the information we know about mass and gravity to make calculations on how objects will react near black holes.