I’m actually just an aspiring web programmer so far with dreams of making video games someday. I have a personal quote: “I’m a programmer, not a mathematician.” I reason out what math I need my program to perform, and only know the math I need to to do that. I don’t need all that excess mathematical stuff that pure mathematicians learn.
I think I’m gonna duck out of intellectual topics for now though. I’ve realized that I have some problems with my thinking that I need to work out.
I think the point he’s making is that, if multiplication is repeated addition, how do you multiply two negative numbers? If 5 x 5 is five plus itself five times, and 5 x -5 is -5 plus itself five times, then what is -5 x -5? -5 plus itself -5 times? How do you have -5 instances of anything? More problematically, what is 5 x pi? How do you have 3.14… infinite non-repeating decimal expansion instances of a thing? Or multiplication of complex numbers, and so on.
Turns out, when you study purer maths, particularly algebra, that we’re not really interested in numbers in the first place. What we wanted all along, in the roll of multiplication, was just some operation on some set that satisfied certain basic properties (like closure, associativity, commutativity, having a zero-element for one of the operations [+], and a distinct 1-element for the other , and so on). Multiplication in the sense of repeated addition satisfied these properties, but then when we generalize mathematics, we keep the properties and let go of this common-sense way of understanding the operation.
we keep the properties and let go of this common-sense way of understanding the operation.
In the process, you let go of all conceptual understanding. So what is the best way to understand -5*-5? You can make an appeal to generalization, but you’ll have to show why the generalization is necessary, valid, appropriate, etc. Even then, that doesn’t serve to help explain it more clearly.
There needs to be a better way of explaining the need or desire to generalize, or of conceptualizing it. Too often teachers simply say, “it turns out…” That leads to a mental dead end, and turns reason into faith.
Only the part of the concept that actually doesn’t matter to what you’re studying. Point is, not all mathematics is meant to count or measure. Complex numbers don’t do this–they, among other things, chart directed line segments. It turns out the way it turns out, because the properties I mentioned above are useful in the situations they’re useful in. For example, applications in number theory and questions like, “Given a tetrahedron, count the number of ways of coloring each face black or white, such that no two instances of the counting could be viewed as the same tetrahedron considered from different perspectives. (E.g. you could color the whole thing white except one face, and you could color the whole thing white except some other face, but they’ll look like exactly the same coloring if in both cases you look at it from the angle of the black face. However, coloring it black on one side cannot be made to look the same as coloring it black on two sides.)” If you replace your domain of numbers with sides of a tetrahedron, and just stick to the requirements of closure, associativity, and so on, you can answer this question by means of an algebra.
As a further note, while you can give some semi-intuitive reasons why multiplying with negatives gives a positive in terms of counting, you can’t give any such understanding for what you get by multiplying irrational numbers.
I think the point he’s making is that, if multiplication is repeated addition, how do you multiply two negative numbers? If 5 x 5 is five plus itself five times, and 5 x -5 is -5 plus itself five times, then what is -5 x -5? -5 plus itself -5 times? How do you have -5 instances of anything?
It means do the opposite of addition five times. 5 x -5 means subtract 5, five times. -5 x -5 means subtract -5 five times.
Honestly if one is going to complain about numbers and math not corresponding to anything “in reality” the problem starts with negative numbers and even zero, not with pi.
The fact of the matter is, electrical engineering (your electrical service) and electronic engineering (your TV, CD/DVD/Blu Ray players, computers, etc.) would never have been developed without the use of complex numbers in the engineering–and those are some “real” number plus an “imaginary” number (dreadful names). Other fields of endeavor use groups of numbers (even groups of complex numbers) laid out in a rectangular pattern, and treat that whole thing as one quantity! It’s all abstract… but if you think about it even the number “1” is an abstraction–it’s up to you to find out something it ties to in reality.
[For the curious, “Real” numbers are the ones that appear on a number line–negative and positive integers, fractions, and “irrational” numbers like pi and the square root of two. “Imaginary” numbers are any of these, multiplied by the square root of minus one, which is written i. So an example of a complex number would be 3 + 2_i_. You can do just about anything with imaginary or even complex numbers you can do with real numbers, even use them in exponents. For example Gauss proved that eπi + 1 = 0. Yes, that’s an irrational real number (2.718281828…), raised to an irrational “imaginary” number, equalling simply -1. Amusing that the five most important numbers in mathematics are all tied together that elegantly, no?]
[For the curious, “Real” numbers are the ones that appear on a number line–negative and positive integers, fractions, and “irrational” numbers like pi and the square root of two. “Imaginary” numbers are any of these, multiplied by the square root of minus one, which is written i. So an example of a complex number would be 3 + 2_i_. You can do just about anything with imaginary or even complex numbers you can do with real numbers, even use them in exponents. For example Gauss proved that eπi + 1 = 0. Yes, that’s an irrational real number (2.718281828…), raised to an irrational “imaginary” number, equalling simply -1. Amusing that the five most important numbers in mathematics are all tied together that elegantly, no?]
Seriously, we’re not children here. I’m fairly certain most people in this discussion have seen that equation by now. Get in line behind everyone else who confuses a discussion on the philosophy of mathematics with a discussion on the science of mathematics.
It’s strange, but I don’t recall anyone having this problem in discussions about incoherent aspects of relativity, or QM. Why is it considered necessary to delve into the conceptual understanding underlying science, but not considered necessary for mathematics?
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?
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.
John Link
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!
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.
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.
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.
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
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.
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.
John Link
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.
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!
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.)
