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# Handling mathematical concepts that have no relation to reality

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Seriously, we're not children here.

I think that part of the difficulty in this thread comes from the fact that the mathematical knowledge of the participants varies greatly, and what some consider to be mathematical concepts unrelated to reality others would be explained by others as lack of understanding of the mathematical concepts.

P.S. By the way, I'm enjoying this thread and the other math thread on .999... tremendously, being reminded of how much I love mathematics. I'm even considering offering my services as a tutor.

P.P.S. I think I must have come across exp(pi*i) + 1 = 0 during graduate school, but I don't remember the proof and maybe I ought to review it. That is really cool!

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If this were a QM discussion, and I said things like, "trust me, the scientists understand it completely", or "at least it's no less incoherent than spacetime curvature in relativity" - both effectively arguments made by aleph_0 in this discussion - I would be considered dishonest or appealing to authority. Why is it acceptable in discussing the philosophy of mathematics?

When I said mathematicians have figured this out, I wasn't making an argument, I was being a little flip. My analysis came from what was above it.

And I'm not sure in what argument I've claimed that something is understood because it's no less understood than something else. I've claimed that we understand complex numbers because we understand real numbers, and I assume that people don't have a problem with real numbers. If that's not true, then we can talk about those, but since we started at complex numbers I think it was a fair assumption that real numbers were not in question.

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P.P.S. I think I must have come across exp(pi*i) + 1 = 0 during graduate school, but I don't remember the proof and maybe I ought to review it. That is really cool!

Well, apparently at least some people here haven't seen the equation--not sure why brian jumped all over me.

Anyhow, John, use the infinite series used to get e^x. plug in i*theta for it. Notice it's the sum of the series for the cosine of x and i times the series for sin x. So e^i*theta = cos theta plus i * sin theta. Now if theta is equal to pi radians.... you get the result that e^(i*pi) = cos(pi) + i sin(pi) = -1.

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In the process, you let go of all conceptual understanding.
Ok, I am completely misunderstanding you or you are making a mountain out of a molehill. Concepts like classical mechanics and the “I before e except after c” might not be 100% accurate, but they do not create problems – concepts are contextual. I still do not see how this is a task for philosophy to solve.

Merry Christmas to everyone!

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And I'm not sure in what argument I've claimed that something is understood because it's no less understood than something else. I've claimed that we understand complex numbers because we understand real numbers, and I assume that people don't have a problem with real numbers. If that's not true, then we can talk about those, but since we started at complex numbers I think it was a fair assumption that real numbers were not in question.

Not quite. The rigorous theory of real numbers was developed from the middle of the 19th century onward. Not all mathematicians were happy about it.

Leopold Kronecker was of the opinion that all of mathematics should be based on the integers. It was he who said "God made the integers. The rest is the work of man". Kronecker did not accept Dedikind's construction of the real numbers (Dedikind Cuts) nor did he accept Cantor's theory of transfinite numbers and the theory of sets. Other mathematicians insisted on strictly finitary constructions, for example Poincare and Brower.

In the 20th and 21st centuries abstract formalism has won out in the market place of theoretical mathematics, but it was not without a fight. Cantor himself was a casualty of the philosophic wars concerning the nature of mathematics. The emnity he faced in academic circles aggravated his tendency to depression and he spent a good part of his adult life in mental institutions. Please see: http://en.wikipedia.org/wiki/Georg_Cantor

Bob Kolker

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I've claimed that we understand complex numbers because we understand real numbers, and I assume that people don't have a problem with real numbers.

That's quite reasonable, but if I remember correctly at least one poster in this thread expressed concern about the validity of negative integers, so the assumption that the real numbers are understood is not going to be satisfied by at least some in this thread.

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That's quite reasonable, but if I remember correctly at least one poster in this thread expressed concern about the validity of negative integers, so the assumption that the real numbers are understood is not going to be satisfied by at least some in this thread.

If it's me you were thinking of--no, that's not what I was saying. I was trying to point out that the logic that makes one reject irrational numbers should make one reject negative numbers (not just integers).

*All* numbers are abstractions, so I see no reason to call only some of them into question. Call them all into question, or none of them.

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If it's me you were thinking of--no, that's not what I was saying. I was trying to point out that the logic that makes one reject irrational numbers should make one reject negative numbers (not just integers).

Agreed. And it seems that there are at least a few posting in this thread who would accept as valid only the positive integers and positive rational numbers. How poor we would be in such a case!

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Yeah, if that is what we were limited to we couldn't call Obama and Huckabee a couple of zeros.

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1^0 = 1

0 sets of 1 apple = 1 apple

I can make no sense of "0 sets of 1 apple = 1 apple".

For now, let's focus on exponentiation on cardinal numbers (whether natural numbers or infinite cardinals):

The key combinatorial fact that motivates taking k^0 as 1 is that, where k and j are cardinal numbers,

k^j is the cardinality of the set of functions from j into k.

Thus, focusing on natural numbers, exponentiation may be defined in at least two equivalent different ways:

(1) Recursively:

k^0 = 1

k^(j+1) = (k^j)*k (where '*' stands for multiplication)

(2) k^j = cardinality of {f | f is a function from j into k}

Considering (2), where j=0 (where zero is taken as the empty set), we have that there is exactly one function from the empty set into any natural number, viz. the empty set itself. (One may object to such a notion of the empty set as a function, but given certain ordinary definitions, the empty set is a function. So, objections on this matter are a different subject from the mere observation that ordinary mathematical combinatorics (whether finite or infinite) takes k^j as the cardinality of the set of functions from j into k.)

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I couldn't understand most of the proofs I've seen for 1 = .999.... But I haven't understood a proof yet that I didn't see a fallacy in.
So, since there are non-fallacious proofs, you've simply not understood one of them. The relevent question then is why you didn't understand them.
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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.

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?

Since it resembles nothing I've ever read in mathematics, why should I be concerned about it at all other than as a gross misstatement of an ordinary mathematical concept? Where did you ever see a text in mathematics state a "mathematical concept" that "2^2 apples is 2 sets of 2 apples" or that "2^0? [is] Zero sets of two apples [which] is one apple"?

When dealing with simple exponentiation on natural numbers, say j and k, a correct and ACTUALLY proffered mathematical concept is that k^j is the number of functions from j into k (where j and k are natural numbers construed to have j and k number of members, respectively (note that is NOT a DEFINITION, thus not introducting a circular definition)).

With your apples, 2^0 apples is 1 apple, since 2^0 = 1, since there is exactly one function from 0 (the empty set) into 2, where 2 is construed to have two members such as does {0 1}.

Or, we may simply take heed of the recursive definition:

k^0 - 1

k^(j+1) = (k^j)*k.

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Thus, focusing on natural numbers, exponentiation may be defined in at least two equivalent different ways:

(1) Recursively:

k^0 = 1

k^(j+1) = (k^j)*k (where '*' stands for multiplication)

(2) k^j = cardinality of {f | f is a function from j into k}

By the way, both of those methods can be justified both in ordinary informal mathematics and in an ordinary set theoretical formalization. However, as an historical note, if I'm not mistaken, it was Godel (using the Chinese Remainder Theorem) who proved that exponentiation can be defined in a system as spare as (what we now call) first order Peano Arithmetic.
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Or, we may simply take heed of the recursive definition:

Let me fix up your typing:

k^(j+1) = (k^j)*k.

let j=-1

k^0 = k/k = 1

Which is what I suggested a few pages ago. All that crap about mapping functions or whatever is waaay too abstract to minimally address this problem.

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Let me fix up your typing:

k^(j+1) = (k^j)*k.

let j=-1

k^0 = k/k = 1

You didn't "fix" my typing, since my typing was correct.

Also, your formulation is not the method of recursive definition on NATURAL numbers, as I specified that my formulation pertains to the NATURAL numbers (extending to the integers in general is another matter).

Again, the standard, and a correct recursive formulation of exponentiation on natural numbers (as can be found in many of thousands of basic texts on the subject) is:

k^0 = 1

k^(j+1) = (k^j)*k.

All that crap about mapping functions or whatever is waaay too abstract to minimally address this problem.
My purpose was not necessarily to "minimally address" anything but rather, as I alluded, to provide a CORRECT understanding of the COMBINATORIAL significance of exponentiation on natural numbers, especially in view of some bizarrely incorrect interpretations that had been posted in this thread. What you refer to as "crap" is quite basic mathematics, correct as I typed it, and serves the aforementioned purpose - understanding exponentiation in a general sense and as a first step toward extending to integers, rational, and reals, as anyone may even better inform himself (past, e.g., "mapping functions" [redundant] or whatever") by looking in the early chapters of many a textbook in undergraduate mathematics.

Please, if there is some MATHEMATICAL error you think you have found in my posts, then I am happy to correct myself or to explain why the content is not correct, as the case may be. But I hope (can't predict) that I would have the restraint to leave unanswered further gratuitously insulting characterizations (such as " crap") of correctly stated mathematics.

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P.S. For that matter, as anyone who has even a smattering of knowledge in just beginning undergraduate combinatory analysis would know, the notion of exponentiation as the number of functions from one set into another provides the basis for the calculation of many even basic problems as applied to probability, odds, and statistics used for everyday physical sciences. So the notion is hardly "waaay to abstract" toward understanding such questions as why k^0 = 1, but is indeed fundamental to understanding a motivation that has practical application beyond merely stipulating that 1 is the value in the base case of some "arbitrary" [my double quote marks] recursive definition.

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The language used is inaccessible, to put the case mildly. You may as well have written it in sanskrit.

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The language used is inaccessible, to put the case mildly. You may as well have written it in sanskrit.

Mathematics is a technical discipline that has a technical vocabulary than can be simplified only so much. May I suggest that if you wish to understand mathematics, you should "learn Sanskrit". The language is NOT inaccessible. It is accessible to those who wish to learn it and take the time and effort to do so. I was able to teach myself calculus at the age of thirteen and set theory at the age of fourteen (that was many years ago). I taught myself mathematical logic at the age of sixteen.

Bob Kolker

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• 6 months later...

Not quite. The rigorous theory of real numbers was developed from the middle of the 19th century onward. Not all mathematicians were happy about it.

Leopold Kronecker was of the opinion that all of mathematics should be based on the integers. It was he who said "God made the integers. The rest is the work of man". Kronecker did not accept Dedikind's construction of the real numbers (Dedikind Cuts) nor did he accept Cantor's theory of transfinite numbers and the theory of sets. Other mathematicians insisted on strictly finitary constructions, for example Poincare and Brower.

In the 20th and 21st centuries abstract formalism has won out in the market place of theoretical mathematics, but it was not without a fight. Cantor himself was a casualty of the philosophic wars concerning the nature of mathematics. The emnity he faced in academic circles aggravated his tendency to depression and he spent a good part of his adult life in mental institutions. Please see: http://en.wikipedia....ki/Georg_Cantor

Bob Kolker

I'm not sure how this suggests that we don't have a firm grasp of what the real numbers are.

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• 3 weeks later...

You guys are making this way too complicated. Mathematics is the science of measurement as I am sure most of you are aware. The purpose of mathematics is to be able to derive values for any measurable aspect of reality , using certain logically derived, objective methods.

Are these methods sometimes highly abstract and beyond an immediate connection to perceptual experience? Sure. Do some methods seem to be arbitrary? One might think that, but if they are logically derived from the basiic axioms of mathematics and allow one to acheive real-world values then they are not arbitrary at all. What matters here is that the mathematical methods used are not complicated beyond neccessity and that they can be used to acheive reasonably reliable and accurate results. Not only intuitive they may be, or how well they relate to every day experience.

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Are these methods sometimes highly abstract and beyond an immediate connection to perceptual experience? Sure. Do some methods seem to be arbitrary? One might think that, but if they are logically derived from the basiic axioms of mathematics and allow one to acheive real-world values then they are not arbitrary at all.

So far, so good.

What matters here is that the mathematical methods used are not complicated beyond neccessity and that they can be used to acheive reasonably reliable and accurate results. Not only intuitive they may be, or how well they relate to every day experience.

And a little more confusing. Reasonably reliable? Reasonably accurate? From a science of measurement?

'Not only intuitive they may be' is really throwing me for a loop.

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So far, so good.

And a little more confusing. Reasonably reliable? Reasonably accurate? From a science of measurement?

'Not only intuitive they may be' is really throwing me for a loop.

Oh forgive me, I typed up that post at like 1am this morning, and it seemed to make sense at the time. Alright...let me try to clear this up.

I mean that the results should be as accurate as possible. But that sometimes given the information available or the methods current known, that the results may be fairly approximate. Possibly due to technical limitations, or the the fact that the mathematical methods known are not quite refined enough to give results that are as precise as one might like. A good example of this is the method of finding the area under a curve by dividing the area in a series of rectangles. This gives an approximate result, however there is a better method, ie integration. But suppose integration had not been discovered, then well the rectangle method might be the best that one could do, even though it wont give exact answers.

The other thing seems to be a messed up sentence fragment...whoops. I beleive that I was attempting to point out some mathematical methods may not be simple to understand intuitively or by resorting to ones everyday experience. Such as imaginary numbers, which have no "physical" equivalent and is purely a method that allows certain types of mathematical problems to be solved. Niether are they a really intuitive concept or one which one can relate to the "physical world" as anything but a method used to measure things in the physical world.

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So the imaginary numbers are utilized as a concept of method, similar to infinity, neither having a physical equivalent, but both providing a utilitarian purpose within math for resolving equations therein.

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• 1 month later...

The modern "concept" of numbers ignores context and conflates distinct concepts. Each class of numbers in modern use is a product of a specific process of accounting. Different accounting context cannot be dropped except in very special cases, e.g., when one is only interested in the quantity and not its units.

Counting leads to counting numbers. The unit is the type of entity counted.

Relating counting numbers leads to rational numbers. The unit is the denominator.

Solving for quantities to fit equations leads to algebraic numbers. The unit is predetermined and then abstracted out of the process -- but the domain of applicability of the equation determines the unit. Complex numbers are solutions of equations, no more, no less -- and labeling the solution to x*x=-1 as "imaginary" is facetious. There is nothing imaginary about the solution to this equation, and physical examples of its reality abound, e.g., electro-magnetics in the context of conductors where phase frequency is complex; or in quantum mechanics where the operators are complex, but the observable eigenvalues are potentially directly measurable, i.e., "real". Complex numbers cannot be directly measured or observed, duh, it would take at least two observations to pin down the real and imaginary components.

Taking the natural limits of sequences of rational numbers leads to the so-called real numbers. The unit is implicit in that the elements of the sequence must be commensurable. This allows one to model even the transcendentals. To my mind, the representation of numbers as limits of sequences, as used in constructing the real numbers, is the most general; but you can't just blithely manipulate sequences as if they were unitary numerals, esp. the non-terminating sequences become bothersome. And you don't get the complex numbers unless you use two sequences.

Thinking of the solution of x*x=-1 as the square root of -1 is just people mistaking formalism for fact. It is just a number defined by the fact that it solves the equation, and should have a disparate name such as "freddie" to make sure folk don't conflate it with observable, measurable quantity.

The problem goes deeper. The notion that -1 or +1 are numbers is false. They have distance and direction. They are vectors. The integers are naturally represented as pairs (x,y) with the equivalence relation (x,y)==(s,t) iff x-y==s-t, with normal vector addition used to construct a vector space from the equivalence classes under this relation across the set of pairs of counting numbers.

So, if -1 is represented as (0,1), and +1 by (1,0), then what sense does it make to take second power roots of either of these vectors?

multiply(-1,-1) = (0,1) * (0,1) = (1,0)

How does that make sense?

The conceptual flaw is that you can't get somewhere by going in the opposite direction, so scaling +1 by multiplying it with -2 has no basis in reality.

Cheers.

- ico

Edited by icosahedron
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The problem goes deeper. The notion that -1 or +1 are numbers is false. They have distance and direction. They are vectors. The integers are naturally represented as pairs (x,y) with the equivalence relation (x,y)==(s,t) iff x-y==s-t, with normal vector addition used to construct a vector space from the equivalence classes under this relation across the set of pairs of counting numbers.

So, if -1 is represented as (0,1), and +1 by (1,0), then what sense does it make to take second power roots of either of these vectors?

multiply(-1,-1) = (0,1) * (0,1) = (1,0)

How does that make sense?

The conceptual flaw is that you can't get somewhere by going in the opposite direction, so scaling +1 by multiplying it with -2 has no basis in reality.

Cheers.

- ico

No, your issue is that you are mistaken in asserting that -1 and 1 have to be treated as two separate dimensions of a vector. You know that in fact, adding +1 and -1 gives you zero. Add your two proposed vectors and you get (1,1) which sure as heck ain't zero. If you make a mistake like this nothing following it will make sense.

If you want to assert that 1 and -1 are vectors (as in some contexts they are) then they are the vectors (1,0) and (-1,0) since the negative sign would denote that -1 has the direction opposite of 1 (i.e., 180 degrees off). It certainly does not lie in the second dimension of the vector, since that would be orthogonal (basically, at a 90 degree angle) to the other dimension.

The second dimension of your vector could be filled by the solution of x*x=-1, which is generally denoted i. So i, not -1, is (0,1). That works well, and is used for the applications you listed above (basically electrical engineering type stuff) whilst complaining about the use of the word "imaginary" to denote such numbers. (The fact is, in mathematics the literal meaning of the word "imaginary" is simply ignored; the numbers are as factual as is π or 2.) Typically though electrical engineers denote such complex numbers with an angle and magnitude rather than using the vector--its equivalent and much more convenient in most contexts.

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