1

This may come across as very confusing, but I'm genuinely confused about the relationship between math as a system humans defined and the "real, physical world."

I feel like when I was in elementary school, we defined arithmetic in terms of physical objects. Let's define addition: if I have 2 apples and you give me 3, I then have 5. We can do the same for subtraction/multiplication/division. Then, we can ask ourselves, "if I have 2 apples, how many apples do you need to takeaway for me to have 0 apples" - we define this as (-2), the additive inverse of 2. But then we need to define arithmetic on negative numbers. If I'm in debt to you 8 apples and in debt to you another 5 apples down the road, then I'm 13 apples in debt to you - this is addition of integers. We can do something similar for subtraction.

But then I get so confused about what to do for multiplication and division since it makes not much sense. Like if I'm in debt to you 8 apples, what does it mean to divide by 2 or -2? Intuitively, I can view it as direction on the real number line (2 negatives = a positive), but this is just pure intuition that has no physical correspondence to the world.

At this point, I wonder whether it's very unhelpful to think in terms of physical objects. So at this point, how do you proceed? Maybe we can use axioms like this answer

enter image description here

Now, we're not using the physical world to guide us anymore, we're using pure math in terms of axioms. And yet, this pure math in terms of axioms coincides with our intuitive understanding of the world (that negative * negative = positive). Is this a coincidence? Like, how is it that our intuition about the world coincides with the proof by pure axioms that negative * negative = positive?

I have no idea if any of this makes sense: I guess the confusion at heart is how we know math represents the physical world, and "what informs what."

If anyone can give me some reading, that'd be helpful too.

On a final note, I don't think my question fully explained my confusion because I'm lost in my thoughts. Any broad comment or reading that doesn't explicitly answer my question is helpful too.

beginner
  • 1,754
  • 5
  • 16
  • 1
    We don't know that math represents anything in the real world (a priori... why should it?). In case we are doing physics (or any other science trying to deal with "the real world") we try to build a mathematical model that fits the experimental data we got. In those cases the math fits the real world by construction (and if it doesn't things tend to become interesting). – Severin Schraven Jan 01 '22 at 06:04
  • 1
    Perhaps you are examining math through a more scientific lens - judging the validity of something based on how well it is supported by empirical evidence. In this instance, this seems flawed to me. Of course, counting as you describe it was created for several different reasons, though I would argue the most important of them is determining relative value of commodities that would be traded between humans (e.g. two cows for one giraffe, etc). If you are wondering whether every aspect of math has such real-world uses, then you need look no further than the vast world of applied math. – Kman3 Jan 01 '22 at 06:05
  • 1
    Who exactly is down-voting this question (and why)? I think that is a question more people should ask themselves. Getting the difference between a model and the thing itself is an important step in becoming a scientist. – Severin Schraven Jan 01 '22 at 06:10
  • @SeverinSchraven I understand the idea that math is comprised of systems/models that humans define. But what doesn't sit so well with me is the source of the model. Like, why are certain things defined the way they are, and why is it that consequences of these things "make sense" / coincide with the real world? E.g. it's a consequence of the axioms used above that neg * neg = positive, but that also intuitively makes sense. I think I'm overthinking this but it's bizarre to realize that I actually have no idea what it means to multiply two neg numbers together despite doing it my whole life. – beginner Jan 01 '22 at 06:16
  • And that is just an example. I feel like if I just knew where "all this started" I'd have a firmer grasp of it all. I've literally been reading the wikipedia pages of "Arithmetic," "Mathematics," etc. – beginner Jan 01 '22 at 06:17
  • We have just defined stuff in such a way that it corresponds to intuition. Then we tried to generalize and the things that worked out are the things that we teach these days. In a sense you are not looking at "the graveyard" of all the things that don't work out, or all the abstract nonesense where our intuition miserably fails. – Severin Schraven Jan 01 '22 at 06:31
  • That's what I thought too ("We have just defined stuff in such a way that it corresponds to intuition") but I see posts all the time on MSE where people say that students should view algebra purely in terms of symbols and axioms, and should not view it in terms of physical objects, intuition. So I'm trying to balance these two competing thoughts. Like, our basis for defs is the physical world and then once we have the definitions, we stop thinking in terms of the physical world? But surely because we "started" with the physical world, any properties we find should "explain" the physical world? – beginner Jan 01 '22 at 06:35
  • Look, in the beginning of mathematics people cared either for applications or for spiritual things. Then we developed the basic notions that evolved over time into the abstract concepts that we have these days. I am not saying that you should abandon intuition (on the contrary!), however, a proof needs to be formulated in this abstract setting (no hand-wave pseudo-arguments about intuitively clear). How do you come up with a proof? Quite often you turn your intuition into an abstract proof (this is what we teach our math students in the first year). – Severin Schraven Jan 01 '22 at 06:41
  • Your question about negative times negative is addressed by this answer. – user21820 Jan 01 '22 at 10:19

0 Answers0