THANK YOU FOR RESPONDING
Technically, it's mildly off topic-ish because I'm presuming realism of some sort, but IDC, I want to talk about this some way!!!!!!!!!!!
The thing is, why do mathematical objects need to be ontologically real? The way I see it is that mathematical objects like numbers or algorythms in scientific theories are descriptions of the physical world a bit like words.
Why would we think that words actually onthologically exists because they accurately describe objects?
I've read that criticism before and I actually agree the standard Quine-Putnam indispensability argument has been refuted. I also think Quine's confirmation holism has been been refuted beyond repair. However, there is a revised version of the indispensability argument, defended best by Alan Baker, which evade all of these criticisms. Baker’s argument avoids them by arguing that mathematics plays more than just a representational role in science, but also plays an explanatory role. This revised argument also helps platonists in the fight against hard road nominalistm too. Although some of our theories can be nominalized, they do so by losing explanatory power.
The argument is analogous to the scientific realist’s argument for quirks and electrons via their explanatory power. Baker’s example comes from biology. One of the puzzles facing biologists is why a certain insect, called the periodical cicada, has prime life cycles. A common explanation is that prime lifecycles cut down on intersection with predators. In this explanation, a property of numbers themselves, that prime numbers have the least multiples, plays an essential part of the explanation and does genuine explanatory work. Just like the scientific realist would argue, you can't have a genuine explanation without the explanans existing.
Sometimes we posit entities that don’t play a causal explanatory role, but a mere logical explanatory role. For example, regardless whether you think this is the best explanation, some use the multiverse to explain the fine-tuning of the universe for intelligent life. It is argued that if there are multiple universes, a fine-tuned universe is really like a win of a cosmic lottery. Notice this explanation is purely logical, there is no causal contact between other universes for this explanation to work. Alan Baker suggests mathematical explanations are like that--they provide logical, not causal explanations.
If mathematical objects were ontologically real, how would that work? For example take a triangle. Say there is an abstract triangle that ontologically exists as an abstract object. In what sense is that a triangle? It is only the concept. If the concept was indeed a triangle itself, it would be a particular and not an ideal. If it is not itself a triangle, how can it be the form of a triangle?
Abstract objects are the nonphysical information encoding of the essence of a thing. We are familiar with information in DNA but the physical stuff isn't the information itself--the information is nonphysical.
For example, when we say sherlock holmes exists, we are talking about Sherlock Holme's properties as encoded information--it's not lke we literally believe there is a detective from England existing that we could go and touch. The same goes with things like numbers and sets.
The whole idea of numbers existing is certainly a
strange idea (probably because everything else we are familiar with are
concrete objects and it's impossible to get a mental image) and one that we should dispense with if we can do it rationally, but it seems we that we can't.
