Conceptual understandings in maths

Maths is founded on axioms, and therefore always arrives at ‘certain’ results, but less frequently at the ‘truth’.

Since the axioms of maths are essentially made up, we can never be sure that they are the ‘truth’, and therefore we can never be sure that the maths branching from these axioms is the ‘truth’, however, we are certain that these results are correct for that system and ‘certain’ for that system.

For example, when doing math problems in class, we are rarely concerned about the ‘truth’ of what we are doing, because we are certain we will arrive at an answer at the end, and that there is a ‘right’ and a ‘wrong’ answer because, of our knowledge that maths is definitive (at least the maths me do). And we also have trust in the credibility of the mathematicians that came up with the maths we learn, for example, Pythagoras’s theorem, or Leibniz and Newton’s calculus. 

I love to ask maths teachers, about the ‘usefulness’ or maths or its ‘applicability’. For example, when I first learned calculus, I was so confused about the use of it, until we came to kinematics in physics. Initially, I understood that if this maths manifests itself in the real world, then it most obviously is true on top of its proof. But this is not necessarily the case, it could be that this maths is currently true and therefore believe the application of it to be a manifestation of its truth as well, but in the future when it is revisited it might be found to be false. For example in the case of Hilbert’s last question, which was ‘proven’ in 1908, only to be visited 81 years later, to find that the proof was incorrect.

A mathematical conjecture has to go through a high standard of logical deductions that build upon each other to get to a complete proof before it can become a theorem. But the initial stage of getting to a conjecture requires a certain level of induction or creativity.

We recently started our unit on mathematical induction, and although the name implies that it should be based around ‘inductive reasoning’, I have come to realise that it is a mixture of both inductive and deductive reasoning or atlas its supposed to. In the induction we have been doing we try to show that a conjecture is true for all positive integers, without having to actually plug in each integer. The process is supposed to be inductive, but to me, it seems deductive in nature because we end up using our understanding of number theory and applying it to our proof. Maybe this is also because we are only handling maths that we know has an answer, so the process does not seem inductive, because we were already told that it works in that way, for example when working with a sequence, we can almost immediately notice whether it is a geometric or arithmetic sequence and because we know what the form of it’s Un formula would look like it does not require as much effort from us. But for actual mathematicians who are dealing with conjectures that are not directly reflected in other maths, there would be an actual process of inductively reasoning a pattern they notice and trying to create a generalisation from it.

Mathematics is a high interdependent area of knowledge, where theories are required to be compatible with each other, within a system because they are based on the same axioms.

Unlike the natural sciences, different mathematical systems are mostly independent of each other, with maybe a few links. In the natural science, there is a quest to find the ‘Theory of Everything’, one single theory that encompasses the entirety of physics and is responsible for explaining the behaviors of micro and the macro of the universe. And just like in maths the two systems, the small stuff delt by quantum physics does not match up with the large stuff delft by general relativity. And this gap is concerning to the science world, whereas if this would be a maths problem it the small and the large would be accepted as two different systems (except that the small stuff makes the large stuff, so it should all link).

The interdependency of maths makes it very fragile, that is why so much focus is put on the rigorous proofs because if one theorem was to later be proved wrong, it would have a ripple effect on the rest of the maths based around it. The biggest mathematical question is the definition of infinity. Mathematicians are torn between adopting two conflicting axioms that both have the potential to push the frontiers of maths, the forcing axioms, and the inner- model axioms. There is a continuum hypothesis which suggests that there are no infinities between integers, but integers can go up to the infinities, and if the forcing axiom is true then the continuum hypothesis is false, and if the inner- model axiom is true then the continuum hypothesis is also true. So although both axioms can help expand our mathematical knowledge in different ways, apparently the forcing axioms are more applicable in the fieldwork, according to Justin Moore (a mathematics professor at Cornell) and he believes that maths should essentially be useful and that’s why forcing axioms are the way to go. This is just an example that shows how it can actually be hard to not contradict in maths, and that the adoption of certain axioms are dependent on what the mathematicians define the purpose of maths to be, establishing it as a human endeavor.— interesting link about the dispute over infinity

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  1. Nick Reply

    This is an outstanding post Aarushi. You have used your own experience and done some great research. Superb. Some thoughts from me:

    1 Your kinematics example, where you say “in the future when it is revisited it might be found to be false”. I guess that’s possible, if exceedingly unlikely. But that’s not the reason we hesitate to use kinematics as a reason to maths is TRUE. The point is that there could be other systems that describe the world equally well; in fact, we know there are. So no one system is true, they just all work. A bit like a Toyota is not the TRUE car. A Nissan works just as well. Perhaps Calculus is a bit like that.

    2 ‘Proof by Indiction’ is actually deductive. It’s confusing – and a language point (irrational* numbers are not irrational#; imaginary* number are real # and complex* numbers are pretty simple# – the point is that the *d words are technical maths words, and the #d words are everyday words. Shouldn’t mix up the two. So *induction is actually #deductive).

    3 I am not sure you have the infinity thing straight in your head. The point is, I think

    aleph-1: the infinity of the integers
    aleph-2: the infinity of the real numbers.

    We know aleph2>aleph1 (Cantors very sexy diagonal proof)
    But there is an aleph between aleph 1 and 2? That is not decidable by the usual axioms, so we need to force it, and introduce it explicitly.

    Happy to explain this last point in more detail at some point in class 🙂


    • Reply

      Oh yeah, I think I wanted to include that point in somewhere, about maths being only one way to explain a system, but it sorta went all wonky when I was writing. But I think I understand now!

      Yes, when I was reading that magazine article I did not completely understand it, but I tried to make sense of whatever I could in here. But yea, wow, I really misunderstood the forcing axiom thing, I thought it was an actual axiom rather than a type/ method of introducing axioms (if that’s the correct way to define it). I would love to understand it better!

      Thank you so much, Mr. Alchin!

  2. Nick Reply

    Book to read over the break for you would be “the Mathematical Experience”

    Brilliant, adn really acccessible

    Happy Holidays!

    • Reply

      Thank you Mr. Alchin!
      Will check it out 🙂
      Also my other post is properly up now, I think it might have gone into the wrong category earlier.

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