Hacker News Reader

Comment by liquidise | Hacker News Reader

Mathematics is a uniquely beautiful field to me. The commutative property has always struck me as special in its own way. 2 x 3 = 3 x 2 feels so obvious, but multiplication is really just addition, and 2 + 2 + 2 = 3 + 3 is far less intuitive, yet states the very same claim.

Most fascinating to me is that many theories are effectively 1-way functions. Entire branches of mathematics have been developed to prove otherwise trivially stated claims. It is something to marvel at, if from a safe distance.

View on HN

The notion that multiplication is ("just") repeated addition is the hardest misconception to overcome blocking progress in mathematics.

Repeated addition is an algorithm that works when you have a non-negative integer argument. Imagining that the algorithm defines the operation limits conception.

I would like to understand what multiplication is in your (and i guess advanced mathematics)

The comment to which you replied says that thinking about multiplication as "just" repeated addition is problematic, so let's look at that.

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.

If we want to define π×π, the best we can do is some kind of algorithm for generating an approximation to π (e.g. as a continued fraction or as a positional decimal fraction) to any desired degree, and then an algorithm for multiplying such approximations. We can prove some bounds on the error introduced by our multiplication algorithm, and that gives us a way of approximating the product to any desired precision.

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.

I don't agree that a definition of multiplication that includes transcendentals such as pi must be numerical. In fact, to the extent that numerical approximations are approximations _of_ something, the thing they would be approximating is the actual value of the operation of multiplication being applied to pi and pi. The only reason we're talking about algorithms at all was to distinguish between them and definitions.

Well, that’s a philosophical rather than mathematical question. I don’t really believe in a concept of “actual value” outside of the context of computations (though I don’t mind conceding it as a matter of convenience and social convention, since the distinction almost never matters for practical purposes). I am not an expert, but mathematicians have investigated this, https://en.wikipedia.org/wiki/Computable_analysis

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.

I am familiar with the constructivist family of ideas. Obviously I agree this a question of philosophy, specifically the philosophy of mathematics. (So not "rather than", since foundations of mathematics is a branch of mathematics.) And because adopting a constructivist approach to mathematics means adopting different ideas about what a mathematical definition _is_ or _can be_, I have to say that it's a bit disingenuous to introduce this context only after engaging in the above discussion.

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.

Easy.

-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

No one is arguing that one cannot implement multiplication algorithms using addition. In fact, one can do it with just NAND. But the definition of multiplication isn't in terms of NAND or addition.

> What about -3 * -2?

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.

The following page explains this one pretty well I think, skip to the "A proof" at the bottom: http://mathforum.org/dr.math/faq/faq.negxneg.html

For that you want to check out Group Theory.

Multiplication can have a number of meanings, but there are two important ones:

- 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.

I had read somewhere that till the point there's only addition, mathematics can remain consistent and complete both. The moment multiplication operation is introduced, Godel's incompleteness theorems come along. Do I understand correctly? If so, what exactly happens when multiplication comes in?

The problem is that there's different types of multiplication. The fact that multiplication over the Reals can be thought of as "iterated addition" is a consequence of commutativity. But mathematicians often generalize multiplication to various contexts as an arbitrarily-defined transformation, which may or may not be commutative. E.g. multiplication over the Reals is commutative; multiplication over Complex Numbers is commutative; multiplication over Quaternions is non-commutative.

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

The "iteration mindset" doesn't always generalize cleanly.

It doesn't, but isn't that, in this case, only because you math gals and guys simply define x^0 to be one? There is no logical reason from a geometry standpoint indeed but you needed a definition.

(x^0) = (1) makes sense because lim{y -> 0} (x^y) = (1).

E.g. (10^0.001) = (1.00230524...)

On-point.

Exponentiation is not like multiplication because it's not associative.

(2^3)^4 != 2^(3^4)

Limited to a numeric sense, multiplication is an operation which began life as repeated addition, and which had to then be extended in various ways as we extended our number systems. We know, for instance, that integral multiplication x * y provides the area of a rectangle with sides of length x, y. This applies without modification to the rationals and reals as well, so we know from this that multiplication remains consistent without having to rely on the concept of repeated addition.

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.

> hardest misconception to overcome

This seems like an exaggeration.

More accurately, it’s a colloquialism.

The number of people who fail to overcome it is close to the number of people who never get a grasp of what mathematics is about. Ask them. It comes up all the time.

Yes, but your inference that having a concept of multiplication as repeated addition leads people to “not grasp mathematics” is backwards.

I have seen little evidence that people who become interested in mathematics have an unusually difficult time “overcoming” this viewpoint.

Their difficulty letting go of repeated addition leads them to consider mathematics opaque, and they lose interest.

I think the big conceptual leap in understanding math is getting beyond the idea that it’s about counting concrete objects in general, and realizing that you can create new sorts of abstract structures that may or may not have anything to do with reality and rules for working with them.

True! I was enlightened in class when we defined addition and multiplication on three random characters then checked if the commutative property holds.

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.

>2 x 3 = 3 x 2 feels so obvious

You just got used to it because you learned it very young. I'm pretty sure it wasn't that obvious when you learned it.

If I remember correctly, in my school multiplication was explained in terms of rectangles made of unit squares. Commutativity was literally visible to the naked eye then.

I also think you define integers as successors. Z, S(Z), S(S(Z)) etc. you can prove that a * b = b * a, given that you have defined the + function and * functions.

It's both intuitively obvious geometrically, and algebraically is conceptually tricky to grasp. Or, more precisely it's trivial to decide that arithmetic operations are commutative or not, but tricky to decide correctly for all operations a child knows.

When my kids started school I was surprised to learn that at age 4 it is not obvious that the number of items on a whiteboard remains constant if all the items are moved to a different part of the board. So what seems obvious, probably is something we just don't remember learning.

2 + 2 + 2 = 3 + 3 becomes much more obvious when you think of a 3x2 rectangle and rotate it by 90 degrees to a 2x3 rectangle.

but then isn't this just a tautological argument? why does multiplying the sides of a rectangle give you the area?

Because of little squares being added.

Multiplication in the form we know it arose from practical needs of area measurement, not as some abstract operation in semirings and rings.

Multiplying the sides gives you the area because multiplication was invented/discovered to do that.

you don't need to invoke a concept of area to justify this. you can just think of a grid of dots. since multiplication is repeated addition, each row represents one step of the repeated addition.

2x3 = 3 + 3

...

3x2 = 2 + 2 + 2

since rotation doesn't change the number of dots, we expect 2x3 and 3x2 grid of dots to contain the same number of dots, which proves that multiplication is commutative.

More interesting, in more general contexts multiplication is not commutative.

For example, if we apply the same pair of 3-dimensional rotations in opposite orders, we generally get different results.

Scaling and planar rotation (in combination, a.k.a. “complex numbers”) are conveniently among the types of commutative multiplication.

> multiplication is really just addition

This is a misleading summary. That multiplication of integers per se can be re-expressed as addition (or if you like, as counting) depends on the basic parts involved being very simple and uniform. The basic multiplication table for integers is just

   × | -1  0  1
  –––––––––––––
  -1 |  1  0 -1
   0 |  0  0  0
   1 | -1  0  1

Since every other integer is just some sum of these basic parts, and multiplication distributes over addition, that covers it. To multiply two integers, first break each one down into some sum of a collection of –1, 0, and 1, then look up each partial product in the basic multiplication table above, and finally sum up all of the results. This process boils down to counting. Since the basic integer multiplication table is commutative, so is integer multiplication in general.

But other kinds of numbers have richer structure based on a richer multiplication table of basic elements.

Well, I am not sure if you've come across some of Bertrand Russell's work, but he qualified Mathematics as having what he called "supreme beauty". The older I get, and I am not a mathematician by any stretch of the imagination, the more I get what he means. This stuff is surely a gift for all of us.

Reply on news.ycombinator.com