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# Handling mathematical concepts that have no relation to reality

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I've been thinking about the philosophy of mathematics lately, and trying to understand how we should handle mathematical concepts that have no coherent connection to reality. Imaginary numbers - for example - pop up all over physics, but obviously never end up being a part of something that we measure. We either measure the coefficient in front of i, or we measure some power of the imaginary number that turns it into a real number. Yet you find people claiming that imaginary numbers are *really out there*, that we are just not capable of detecting them, or making sense of them.

Leaving aside imaginary numbers, how about simple things like negative numbers. The justification is, "draw a line, mark the center as zero, and if numbers to the right are positive, then numbers to the left are negative." I would simply say, "why don't you just set the left-most number as zero?" Obviously negative numbers can serve as direction indicators, but why not just directly refer to those directions? So e.g. if I'm trying to figure out how much I would move forward after taking two steps forward and one step backward, I don't say, "I'm taking two steps, and one negative step." The problem is understood to have a direction component.

Or even more simple, take exponents. 2^2 apples is 2 sets of 2 apples. Well, that's 4 apples. 1^3 apples is one set of (one set of one apple)) which is just one apple. But now what about 2^0? Zero sets of two apples? Well, that is one apple. And 2^-1? A negative set of two apples? Well, that is half an apple.

Now this makes absolutely no sense and has no relation to reality. So how do we as philosophers go about handling it? How do we reconcile the need for concepts to be tied to reality with the existence of mathematical concepts that have no tie to reality?

Some might claim that once we specify a few connections to reality - a few foundational principles - that we can then ignore reality and go off on our own mathematical adventure, wherever it leads us. And that has worked. Whenever mathematics is tested against reality - i.e. whenever the equations are solved in a way that produces something measurable - they are correct. But what is the justification for this claim that only a few concepts need be tied to reality, and what of the other concepts that have no tie to reality? In what sense are they real?

And what about those who look at certain physics problems - e.g. a fundamental theory of everything -, find certain patterns that lead them to use certain sets of equations - e.g. equations for the mechanical oscillation of a string -, and then form conclusions that are incoherent but measurably correct - e.g. string theory. Where does the pitfall lie - in our ability to grasp the theory, or in the theory's ability to explain reality to us? In what sense is the theory true, and how do you go about definitively saying that it is true?

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Obviously there is no such quantity as -1 in real life, and so obviously you can't find the square root of a non-existent quantity. But things like negative numbers and imaginary numbers are concepts of method. So they still have use in relation to quantities that do exist. You can understand the concept of -1 because you understand positive numbers and 0. On, say, the Cartesian plane, it would get pretty confusing if you just used positive numbers for every quadrant. You need some method for making that easier, and that's where negative numbers can help; the negative of course just indicating direction.

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Obviously there is no such quantity as -1 in real life

But of course there is! If my liabilities exceed my assets by one dollar then my net worth is -\$1.

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Now this makes absolutely no sense and has no relation to reality.
This is the part that I don't understand. In what way does "-1" make absolutely no sense and have no relation to reality? As an alternative, in what way does 434,765,544,376 make any sense and have any relation to reality (or, would you say that in fact it doesn't)? Please clarify.
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But of course there is! If my liabilities exceed my assets by one dollar then my net worth is -\$1.

Edited by dream_weaver
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This is the part that I don't understand. In what way does "-1" make absolutely no sense and have no relation to reality?

I was referring to the paragraph that immediately preceded the quote, which was about exponents less than one, not about negative numbers.

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But of course there is! If my liabilities exceed my assets by one dollar then my net worth is -\$1.

Hmm, I just need to clarify my point. Your example is true, but that's not what I meant. For instance, a one dollar bill physically exists. It is an entity. And there is one of those bills. But you cannot have -1 of those bills. Do you understand what I mean? The least quantity any entity can be is 1. If an entity has 0 quantity, then it does not exist. Therefore, in this sense, negative quantities are an impossibility and would imply some alternative to existence vs. non-existence, like...less than non-existence.

Even when your liabilities exceed your assets, the fact that your net worth is a negative number is not an actual entity. This is still a concept of method. Physically what is going on is you lose all your assets, but you don't continue losing assets once you go negative. You have no more assets to lose when you have 0 assets! No, what happens is you go in debt. You don't have anymore assets, so you are going to have to get more in the future. I hope my point is clear.

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I'm trying to tie the notion of 0 exponential to entities in reality (such as apples) in some understandable way. Why is it that integers with integer exponents greater than 0 work perfectly fine for counting entities (e.g. 2^2 is 2 sets of 2 apples), but as soon as that integer exponent goes lower than 1, it becomes useless for counting?

1^0 = 1

0 sets of 1 apple = 1 apple

Edited by brian0918
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I was referring to the paragraph that immediately preceded the quote, which was about exponents less than one, not about negative numbers.
Fine, so substitute exponents less than one. If you don't have a problem with negative numbers, then maybe you can explain in terms of why integers are okay, versus less-than-1 exponents are not. I'm asking for more detail, because I don't see why it causes you problems.
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Fine, so substitute exponents less than one. If you don't have a problem with negative numbers, then maybe you can explain in terms of why integers are okay, versus less-than-1 exponents are not. I'm asking for more detail, because I don't see why it causes you problems.

Why is it that integers with integer exponents greater than 0 work perfectly fine for counting entities (e.g. 2^2 is 2 sets of 2 apples), but as soon as that integer exponent goes lower than 1, it becomes useless for counting?
Edited by brian0918
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I'm trying to tie the notion of 0 exponential to entities in reality (such as apples) in some understandable way. Why is it that integers with integer exponents greater than 0 work perfectly fine for counting entities (e.g. 2^2 is 2 sets of 2 apples), but as soon as that integer exponent goes lower than 1, it becomes divorced from reality and useless for counting?
Fine, be that way, answer my question 2 minutes before I ask it 2^2.5 apples is 5.65685425 apples -- I assume that's okay. Then 2^.5 applies is 1.41421356 apples and 2^-.5 applies is 0.707106781 apple. No matter how hard I try, I can't even get actual zero apple using this method. What is it that doesn't make sense? Unless...

0 sets of 1 apple = 1 apple
No, that isn't what exponentiation means. There is no simple ordinary English concept that translates exponentiation, not as far as I know.
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No, that isn't what exponentiation means. There is no simple ordinary English concept that translates exponentiation, not as far as I know.

Well it works perfectly fine in that way for any integer exponential above 0. 2^1 is 1 set of 2 apples. 2^3 is two sets of (two sets of two apples). Why does this breakdown in explicability occur as soon as the integer exponent drops to 0 and/or goes negative?

Edited by brian0918
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The need for this discussion is due to the way in which math is taught to kids. They're given positive integer exponents, and told to relate it back conceptually to counting apples. Then one day the teacher throws in decimal exponents or exponents less than 1, completely destroying that conceptual foundation.

There has to be a better way to teach that, and other abstract concepts such as imaginary numbers. Otherwise, you end up with people claiming that the complex number system is somehow physically real - that we should be able to count imaginary components of apples -, but that our minds are simply incapable of grasping or detecting them.

Edited by brian0918
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There has to be a better way to teach that, and other abstract concepts such as imaginary numbers.

You could look into how these subjects are taught in the Montessori or Mortensen systems. These are both systems that teach math using physical materials that kids are taught to manipulate so that they learn math through experience. I'm not completely familiar with them but it might be something for you to look into since you're interested.

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Thanks for the suggestions Heather, I'll check them out.

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The need for this discussion is due to the way in which math is taught to kids. They're given positive integer exponents, and told to relate it back conceptually to counting apples. Then one day the teacher throws in decimal exponents or exponents less than 1, completely destroying that conceptual foundation.
Okay; then, I agree. I know that it is possible to teach the concept of exponentiation to at least 1 child around age 8 in a way that avoids the lie, but that's not generally how it's done.
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Complex numbers have a geometric interpretation; they are, with the exception of definitions of derivation and continuity and the like, just pairs of real numbers. Don't worry about them, we have this thing figured out.

Take a slightly more abstract mathematical concept, that of isomorphism. This actually corresponds to nothing at all, and is--in a sense--a way of reading off mathematical structure imposed on a set. (That's not really what it is, but that's what it's used for, so close enough.) So sometimes our mathematical structures are just devised in order to organize our other mathematical structures. This kind of phenomenon, that of mathematics for the purpose of mathematics (not in the Platonic sense) comes up a lot in foundations. These are not troubling concepts, in my opinion.

In the philosophy of mathematics, I think there are a few related, seminal questions: What is a mathematical entity, what is mathematics as a whole, how does mathematics figure in explanations, and how do we bridge the gap between the physical world and it (i.e. problems of model theory)? These have, by far, much harder answers.

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I've been thinking about the philosophy of mathematics lately, and trying to understand how we should handle mathematical concepts that have no coherent connection to reality. Imaginary numbers - for example - pop up all over physics, but obviously never end up being a part of something that we measure.

Imaginary numbers are a mathematical tool, like a vector or a matrix. A matrix is not something that you measure either - it's a way of presentation.

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At least from what I know, they are used to represent a 2 dimensional vector in physics containing information about both the phase and the amplitude of something (like current or voltage).

In some equations using square root of -1 as a "number" helps get solutions to the equations, but only if one knows how to translate the result to the real physical world. Let's see if I can remember...

Every imaginary number can be translated into a vector (in 2D). Ordinary numbers, without the complex part, are actually a complex number with the imaginary part being zero.

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Basically, think to yourself: Suppose you have the equation X^2= -1.

What does this equation mean? Where did it come from? What is it's source in reality. It has to have one, otherwise I can write 0=1 and say "ok guys, let's discuss this, it's a serious problem that needs solution" - obviously one can't just write whatever one wants - the problem has to come from reality. OK then, so what about the equation above?

The equation only makes sense if you have a vector that can be represented as an arrow on a circle. What kind of entity would require such representation? A voltage that is changing over time, for example. At a given moment in time, in a specific location in a circuit, the voltage has a certain "phase" - a certain shift in time compared to the starting point of measurement. It's important to keep track of that shift, as well as the amplitude. So one can keep track of the phase by keeping track of where on the circle the "dial" of the voltage is (12 O'clock, 6 O'clock etc'). The way to do it is by using a real axis and an imaginary one; then the voltage can be described by two components:

cos (angle) +/- i sin(angle)

(multiplied by the amplitude).

Then, when you use this form of representation in the equation, you get the result of i or -i which means that the "dial" is on 12 O'clock or 6 O'clock in your circle.

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Exponents have rules for manipulating them when in ratios. For example:

22/23 = 2*2/2*2*2 = 2(2-3) = 2-1 = 1/2 This is rapid cancellation of factors. It works because the notation it is manipulating is valid.

And for the case when the ratio is equal:

22/22 = 2*2/2*2 = 2(2-2) = 20 = 1

Multiplication is repeated addition, but exponentiation is repeated multiplication. Exponentiation is a method built on a method, not built on units directly as addition is.

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Imaginary numbers are a mathematical tool, like a vector or a matrix. A matrix is not something that you measure either - it's a way of presentation.

Right, but my concern comes from the fact that people think these things *are* physical, actually part of the universe, and that we are simply incapable of perceiving or understanding them. If these concepts were more easily relateable to the real world, or put in the proper context, these confusions wouldn't occur.

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Basically, think to yourself: Suppose you have the equation X^2= -1.

What does this equation mean? Where did it come from? What is it's source in reality. It has to have one, otherwise I can write 0=1 and say "ok guys, let's discuss this, it's a serious problem that needs solution" - obviously one can't just write whatever one wants - the problem has to come from reality. OK then, so what about the equation above?

The equation only makes sense if you have a vector that can be represented as an arrow on a circle. What kind of entity would require such representation? A voltage that is changing over time, for example. At a given moment in time, in a specific location in a circuit, the voltage has a certain "phase" - a certain shift in time compared to the starting point of measurement. It's important to keep track of that shift, as well as the amplitude. So one can keep track of the phase by keeping track of where on the circle the "dial" of the voltage is (12 O'clock, 6 O'clock etc'). The way to do it is by using a real axis and an imaginary one; then the voltage can be described by two components:

cos (angle) +/- i sin(angle)

(multiplied by the amplitude).

Then, when you use this form of representation in the equation, you get the result of i or -i which means that the "dial" is on 12 O'clock or 6 O'clock in your circle.

Correct. I dealt with all of that in my circuits courses, and that is definitely the best way to understand imaginary numbers and tie them to reality. Unfortunately, kids are taught about imaginary numbers without this connection to reality being made, and given the impression that the numbers are somehow magical, or that we are simply incapable of grasping their true nature.

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Brian, Here's how I might relate various exponents to reality:

Exponents can be used to describe cell division. Starting with x cells, the number of cells after any number of divisions can be calculated by x · 2^n, where n is the number of divisions that have taken place.

Say you have x cells and you want to know how many cells there were n divisions ago. You can calculate this by x · 2^-n. So negative integer exponents are instructions for sucessive division in the same way that positive exponents are instructions for successive multiplication. In order for the process to be continuous and mathematically sensible, you must define x^0 = 1 (the multiplicative identity). This too can be related to reality though: how many cells are there after 0 divisions (i.e. now)... x · 2^0 = x · 1 = x.

Now, say you didn't know that the number of cells doubled with each division. The only information you have is that you started with w cells and ended with x cells after n divisions. The cellular multiplication per division can be calculated by (x/w)^(1/n).

That covers positive, zero, negative, and (simple) rational exponents. As one's math training advances you can discuss formalities/identities like a^(m+n) = a^m · a^n and more complex concepts like irrational exponents (these are tough). It's been a while since I studied it, but I believe irrational exponents are only truly calculable through series expansion.

Does that help?

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That covers positive, zero, negative, and (simple) rational exponents. As one's math training advances you can discuss formalities/identities like a^(m+n) = a^m · a^n and more complex concepts like irrational exponents (these are tough). It's been a while since I studied it, but I believe irrational exponents are only truly calculable through series expansion.

Does that help?

Or as the limit of a sequence. Let the sequence {a\sub n} converge to a, where a\sub n is rational. Then we can write b^a = lim (n -> inf) b^a\sub n. Since each a\sub n can be written in the form M/N (we assume a\sub n is rational) the b^(M/N) is just the N-th root of b^M a well defined quantity. The inverse to exponentiation is the logarithm which can be defined by a convergent series for any real positive value. [We are assuming the system of real numbers. For complex numbers this definition can be extended]

Bob Kolker

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