Repeated addition is an algorithm that works when you have a non-negative integer argument. Imagining that the algorithm defines the operation limits conception.
Turned out you can do whatever you want... It just that usually you want addition to be commutative and usually want to use an infinite set of symbols called numbers.
This seems like an exaggeration.
I have seen little evidence that people who become interested in mathematics have an unusually difficult time “overcoming” this viewpoint.
Consider 3 x 2. If we take that approach, it seems ok - we understand it to mean "add together 3 2's" - 2 + 2 + 2, which gives the correct answer of 6.
What about -3 * -2? What does it mean to add a negative number of times?
What about pi * pi? What does it mean to add something pi times?
What about the matrix M * the matrix N?
etc. The parent's point is that this repeated addition thing is just an algorithm one can use to calculate a multiplication for some operands, specifically whole numbers, not a general definition of the operation of multiplication.
For example if we want to know π^2 to about 3 decimal digits (~9.87), we can start by using an approximation of π to about 4 decimal places (~3.142) and then multiplying the two decimals.
For rational numbers, multiplication algorithms are usually built on breaking a number down into constituent pieces, multiplying every pair of pieces from the two multiplicands, and then adding up all of the partial products.
Matrix multiplication has the additional complication that the elementary terms involved (entries in different places in the matrix) cannot be added to each-other. But the basic procedure is still the same: break the two multiplicands down into basic units which we already know a multiplication table for, compute all of the partial products, then sum them up.
We can treat π purely symbolically if we like, but as soon as we want to do anything useful with it we need some kind of approximation or algorithm.
For example, if someone asks you to explain Euclid's proof of the infinitude of the primes, and you say that Euclid did not provide any such proof and nothing more, I think it's quite disingenuous. It would be more proper to say, from a constructivist view, the argument Euclid made isn't a valid proof, and then either explain the proof in the logical context in which it was made or decline to.
In this case, the point of discussion was separating the definition of multiplication from an algorithm implementing it. It's quite unfair to silently take a position that a mathematical definition without an algorithm isn't valid or meaningful and then on that basis argue that only numerical approximations to transcendentals have meaning.
So many common mathematical concepts such as "the integers" have no meaning in a constructivist approach that it's not sensible to engage in mathematical discussion without establishing that one's fundamental basis of approach varies so widely from the common one.
-3 x -2, multiply signs first: +, remainder: 3x2 repeat 2 3 times and add; 2 + 2 + 2 = 3, remainder: 0
3.141 * 3.141, multiply signs first: +, remainder 3x2 repeat 3.141 3 times and add; 3.141 + 3.141 + 3.141 = 9,432, remainder: 3.141 * 0.141
shift decimal: 3.141 * 1.41, remainder: 3.141 * 1.41
3.141 * 1.41, repeat 3.141 1 times and add: 3.141 = 3.141, remainder: 3.141 * 0.41
shift onto result: 9.432 + 0.3141 = 9.7461, remainder: 3.141 * 0.41
shift decimal: 3.141 * 4.1
repeat 3.141 4 times and add; 12.564, remainder: 3.1410.1
shift onto result: 9.87174
shift decimal: 3.141 1
shift onto result: 10,053764
result: 9,875881
check back with calculator: 3.141 * 3.141 = 9,875881
Multiplication can be represented as a number of additions and shifting the results of substeps over the decimal point
My son actually asked me that a while back (or rather, “why does a negative times a negative make a positive?”), and I honestly didn’t have a very satisfying answer. The best I could come up with was to go back to the definition of multiplication as repeated addition and then start with multiplying/repeated-adding a negative number by a positive number: that would work backward on the number line and give you a negative number. Then, since multiplication is commutative, a positive number multiplied by a negative number must behave the same way: multiplying a positive by a negative causes the negative to move backwards. So, if multiplying a positive by a negative moves it the “other way”, multiplying a negative by a negative must move the first number to the right.
It works, but it’s not as intuitive as I would have liked - I did better with why negative exponents are 1/x^n and why the angles in a triangle add up to 90 degrees.
- multiplication as function composition, eg the product of two matrices is the linear transform obtained by applying the right matrix and then the left (onto a column vector)
- multiplication as “the operation that distributes over addition” — in the theory of rings and algebras, if you have one operator “+” which forms an abelian group, then any operator “•” such that (a+b)•(c+d) = ac+ad+bc+bd for all a,b,c,d then you have a ring (if • is associative) or an algebra (otherwise) and you can apply all kinds of structure theorems.
This is very oversimplified but it’s important to remember that you can multiply lots of things which don’t look like integers at all.
Incidentally, "exponentiation is just repeated multiplication", right? Try using iteration to solve 0^0.
lim{x -> 0} (0^x) = 0
lim{x -> 0} (x^0) = 1
E.g. (10^0.001) = (1.00230524...)
Exponentiation is not like multiplication because it's not associative.
(2^3)^4 != 2^(3^4)
Similar extensions happened with negative arguments. At a certain point, pure intuition vanishes, and instead you are left with the task of simply creating a set of rules that define a consistent mathematics. Often times, the approach you take is limited by this very requirement; our mathematics is only consistent if a negative multiplied by a negative results in a positive, for instance.
This topic hints at the gap between applied mathematics rooted in concrete reality, as it were, and theoretical mathematics, which of course deals purely with abstract structures.