It is easy to prove for finite sets (just count) but much harder for infinite ones. For example, which has more subsets, the integers or the positive integers? How about the integers and the reals?
If you answered "the integers" to the first question, you aren't thinking about this right, as the integers and the positive integers have the same number of elements to start with. source: https://en.wikipedia.org/wiki/Aleph_number
Another example of why mathematics are a wrong abstraction to be optimally useful.
Mathematics should have bounds in the same way as our universe has bounds.
Any theorem that has a different behavior if something is infinite doesn't matter at all and is a waste of time for real engineers who solve things in the real world. The niche of mathematics that describe things beyond what our universe has to offer should be an optional extension to the mathematical framework instead of being the default, in order to not pollute discussions.
This is a fine and defensible (although not mainstream) viewpoint with several adherents among mathematicians and various levels of success in making it formal and precise.
I don't think finitists have a problem with convergent sequences, they just wouldn't talk about infinities, rather than extending the sequence gets you as arbitrarily to 1/3 as you wish to go
Since you feel so strongly about it, let me argue that you're simply wrong:
Infinities have the opposite effect then you might think, they make things simpler. It's much easier to reason about an infinite list of numbers then to reason about 64 bit numbers.
In analysis, it's much easier to reason about infinitely differentiable, or smooth, surfaces then very rigid and complicated services.
The fact that infinities cause some complications in the foundations itself is a drop in the bucket to the practical application and simplifications of day to day mathematics.
> Infinities have the opposite effect then you might think, they make things simpler. It's much easier to reason about an infinite list of numbers then to reason about 64 bit numbers.
I would argue that a few extra lines in a proof is a small price to pay to avoid the Godelian catastrophe.
A formalist might say that mathematics has no infinities. Even infinite sets are just definitions of finite length, even if expanded all the way to ZFC axioms. The problem is your intuitive understanding.
A realist might say that mathematics exists within this universe, therefore it is equally subject to its constraints just like anything else. Problems arise via misapplication or misinterpretation of the math.
A model theorist might point out that even in models of math with only a countable collection of objects, it's still true that the reals are uncountable. The problem isn't with infinitie, it's that the finitary objects have subtle interactions.
A logician might object that ZFC already does start of with an intuitive notion of infinity, i.e. the counting numbers is a natural collection. The problem is that this inevitably has unintended consequences.
A historian might gently point out that this debate already occurred vigorously about 150 years ago, and the verdict was that we just gotta live with the weirdness of infinities. The problem is that we lose too much useful math by trying to throw them out.
Yes, my current inclination would be to define truth (in a practical sense) as that which is computable. Everything else belongs to the realm of fiction or fantasy.
Note that I don't mean to imply that this is the entirety of my philosophy. Just that one should distinguish between what is practically relevant and what is not and I think the current state of mathematics and CS does a poor job of making this distinction.
The most prolific contemporary proponent of this view is probably Norman Wildberger, who takes a very unconventional approach and tries to build everything without real numbers, which he doesn't believe make physical or logical sense and therefore shouldn't be used as a basis for anything.
Calculus would likely not be invented without infinity. Many theorems, identities, and techniques may be approximated but ultimately rely on proofs using infinities - I doubt we would have discovered them quickly or at all without infinity. After all, the very concept of a limit evokes the concept of an infinite sequence.
That is true historically but I doubt that it is true necessarily. I mean I suspect that starting from what we now know it should be possible to reconstruct calculus (at least for all practical purposes) without reference to infinities or infinitesimals.
One argument in favor of this belief is that neither practical computations nor analytic intuition require actual infinitesimals.
The later, at least, has been my experience. I think I have a fairly decent practical intuition for calculus based on imagining dx becoming smaller and smaller until it's small enough, but I don't think my brain has any actual representation of "true infinitesimals" and my intuition breaks down completely if I try to imagine things like the relationship between the rational and irrational numbers. Maybe that's due to my intellectual limitations, but I wonder if it isn't because these concepts might be over-elaborate abstractions that don't really exist in our world.
If you answered "the integers" to the first question, you aren't thinking about this right, as the integers and the positive integers have the same number of elements to start with. source: https://en.wikipedia.org/wiki/Aleph_number