Empty Set doesn't exist

If you cant sense it, then it doesnt exist.

You cant see emptiness, therefore emptiness doesnt exist.

Do not argue with the ultrafinist, or WWBT! (I should know better, for I'm about to)

He doesn't even believe in the concept of natural numbers, much less in any more complex abstract concepts (such as set theories).
The only thing he seems to believe in are his direct senses, but he refuses to actually infer (by induction or deduction) anything from them.
He claims mathematics is religion, and I'll partially agree with him on this(more on this soon), although mathematics just teaches "If A then B" and it doesn't force you to believe in some particular A. Mathematics proves very useful in the real world and thus the consequences and things it teaches have a good chance of being true. There is another way to look at this: claim that computation exists (in the Turing-equivalent sense), and you also get most mathematical theorems from that. The thing about computation is that it just means that simple finite abstract rules can be followed and will always give the same result in all possible worlds. It can be shown that computation can be encoded in arithmetic and a Turing Machine can also do arithmetic (unbounded). The Chuch-Turing Thesis shows that computation is universal and could even be considered a natural class. If you posit a reality having well-defined physical laws, you'll also have to posit some form of computation (our world is capable of it, at least in the Finite State Machine sense, a restricted Turing Machine). If after some thought experiments you end up subscribing to functionalism or computationalism, you also have to put computation in the ontology (actually it's the only thing that you truly require). Such beliefs, be they in a reality, your own mind or in the fact that computation works, or in mathematical truths are "religious" beliefs, although they do have substatial evidence for them being true - in another way, some parts of math are good things to bet on being true. If you ask me a belief in the Chuch Turing Thesis is a good "religious" belief - a bet on something being true(even if supported by evidence), such a bet taken more seriously can lead you to some forms of Platonism (again, religious belief). Some beliefs are incompatible, for example, the belief in functionalism and a negative belief in computation, or a belief in an ontologically primitive reality and computationalism (the version where one assumes one's mind exists), and one should not hold incompatible beliefs if they want to be correct.

Me, I have no problem in making such bets (about certain things being true) when I have good evidence for them. Why? I risk being more wrong and stand to gain getting closer to more true statements. Of course, a smarter way to go about this is to do uncertain inference: hold confidence values for each belief and adjust them based on evidence.

Too bad our local ultrafinitist doesn't want to have any beliefs, and so he will never ever have a chance of being right or wrong. He may hold various beliefs, but will refuse to acknowledge that he holds them. In other cases, he must bet on some theory, and if he bets or not is itself a choice, and not betting is also a form of "belief" (or lack of).

Good luck to you, ``in LISP''/symta-guy, and I hope you do some more thinking about what beliefs are acceptable to you and which are not, as well as what standards you use to select them. It's easy to pick some ultraskeptical standard which doesn't allow you to hold any beliefs at all, but that limits greatly one's abilities, to the point of never being able to be right or wrong, or making any choices. I think it's great having a chance of being right, and if you're wrong, that's not a problem as long as you retain the ability of recognizing wrong beliefs and removing/replacing them, thus being less wrong as time passes (and now I expect that you'll dispute time's existence - that can be done as well, but again limits your options!).

>>58
I just want the inference to be justified by senses too.
The problem with this is that you won't be able to form any theories/metatheories about why you have senses at all, or why physics (or just whatever makes your senses be this or that) behaves the way it does and so on.

That inductive proposition talks about abstract systems, in the real-world, you will have a local limit imposed by physics.

If for example you deduced by induction that reality corresponds to some specific mathematical structure. Why only that structure? Let's say its information limit is some constant k. Why not k-1 or k+1? Occam's Razor suggests that the simplest theory is that all possibilities are realizied, thus also the k+1 world. A belief in only world k is a stronger belief than a belief in all finite worlds 0,1,...,k,k+1,...
It's a stronger belief just like the belief that the sun doesn't exist as long as you don't look at it(such as at night).

Math itself tries to be as general as possible - in that it can also be used by physics, but general enough to not limit itself to only what's locally physically possible. There is no reason why any finite natural number should be the limit, hence why you get a countable infinity of natural numbers.

You could say that all this is abstract non-sense, and that you don't care about why anything behaves like it does.
Consider this then: in the far future, you're very old and have developed some incurable brain tumor, the doctor offers you to get a digital brain replacement (be it gradual or instant, this is a thought experiment, so pick whatever suits you best), do you say yes or no? You have to bet on a theory because your future experiences depend on it, yet without having worked out all the possibilities you cannot make a choice, but you don't have the luxury of not making a choice - a default no also has costs.