Thread · HN Continuity

I'm pretty sure the vast majority of people understand linear algebra etc, if they were taught that the reason why linear algebra works is that algebra (algebra over a field) are an approximate isomorphism to physical reality.

From that perspective you do not need contrived examples like using function composition as alternative operator.

It would be simply enough to teach the student that any symbol system following the definition of an algebra over a field has the same properties as e.g. natural numbers with respect to the (possibly physical) composition operator.

If you don't understand this then it's because you don't understand linear algebra.

Consider this, you can write numbers as symbols or as words. There is no single universal language so a number can be represented by many words and mean the same thing.

Now let's ignore the concept of a number to begin with. Let's say numbers were never invented.

Everyone discovered that you can combine symbols a+a=b b+a=c and so on. They realize that the laws relating to how the words can be combined follow certain properties.

They will also notice that marbles under physical composition follow the same rules.

They then discover that the effect of having a marble in one hand and in another can be physically composed into one hand having two marbles.

They then think about giving the marble quantities a different representation, e.g. one marble = 1 finger

They invented an isomorphism.

Transforming the marbles into fingers then adding the fingers and then transforming back to marbles is the same as taking one and one marbles and putting them together.

And that is how linear algebra was invented, long before anyone even know what a mathematical definition was.

Edit: if it's not clear, any physical composition operator can form an algebra over a field,

In a mechanical calculator, the gear movement is the composition operator

In a computer the adder is an electron based composition operator

Hence the reason why it is possible to build useful computers that perform useful calculations relates directly to the fact that there is an isomorphism between the real world state and the computer state

Sorry, long rant incoming. I think there are a lot of things wrong here.

I think you are right where you talk about the "one marble" and "one finger" isomorphism, that makes sense.

Everywhere else, you throw around a lot of terminology that gets very confusing.

> any symbol system following the definition of an algebra over a field has the same properties as e.g. natural numbers with respect to the (possibly physical) composition operator

What do you mean by symbol system? Are you suddenly in the discipline of logic and formal languages here? What are "natural numbers with respect to the composition operator"? what is _the_ composition operator? Function composition? I don't see any K-algebra structure here, please enlighten me. What does it mean to be physical here, referring to the composition operator? An algebra over a field is a pretty fundamental structure, intuitively polynomials over a field with some extra rules. But this is all still very abstract and I don't see how it is immediately apparent to be an "approximate isomorphism to physical reality", which I can only assume to mean what I think it means.

> Let's say numbers were never invented. [...] Everyone discovered that you can combine symbols a+a=b b+a=c and so on.

Whoah, slow down there, you are already assuming a lot. Infix notation, equality, and suggestively using the plus symbol. I would agree with your claim "everyone discovered [...]" when talking about concepts like addition, but you are talking about abstract "combining symbols", which in the way we do it now is a rather modern advancement, especially because you talk about the logic/language theory concept of words.

> Transforming the marbles into fingers then adding the fingers and then transforming back to marbles is the same as taking one and one marbles and putting them together. And that is how linear algebra was invented, long before anyone even know what a mathematical definition was

That is not linear algebra. That is simply group or ring theory, and really only restricted to the integers, really. The integers (or whatever you mean by "linear algebra") were exactly definitionally invented the moment they had a kind of mathematical definition of some sort. I think you are trying to say the concepts were discovered and/or used long before anyone knew about a mathematical definition.

> if it's not clear, any physical composition operator can form an algebra over a field

Please explain this better, I am not sure what you mean by a "physical composition operator". Do you mean that functions that are already K-linear operators over vector spaces form a (non commutative) K-algebra with the function composition operator as multiplication? Sure, that is true, but that already assumes so much structure. I don't think it is clear at all. I am not saying you are wrong here (smart physics concepts stuff around Noether theorem etc. come to mind, but I am not sure), but I don't think this is easy to see intuitively at all, as opposed to marbles and fingers etc.

> In a mechanical calculator, the gear movement is the composition operator. In a computer the adder is an electron based composition operator

Whoah, slow down again, please. What composition operator? There is no linear algebra here. Also, it is all over the integers, if anything, not over a field. Or if it is a field, then it is just approximated with floats, which mathematically amounts to things that are not even a true ring/algebra, because of rounding errors.

> Hence the reason why it is possible to build useful computers that perform useful calculations relates directly to the fact that there is an isomorphism between the real world state and the computer state

You make the conclusion sound so easy, yet you left out all of the important steps. And computers are famously not there as an "isomorphism between the real world state and the computer state", if you talk about physical stuff, as they can only approximate. If you are talking about fingers and marbles, then sure, I guess the computer reflects that.

However, the marbles vs fingers vs the abstract notion of a number have a totally, completely different flavor than physics and linear algebra. The classic undergrad philosophical debate around "are numbers reality?" exactly explains the fact about how the concept of counting is already an abstraction, because we don't care about the physics of the atoms in the marbles or the fingers or whatever. With physics, the separation of math vs physics, or abstraction vs reality, is a bit clearer, because we use math to model/describe/approximate reality as reality unfolds etc.