Comment by trabant00 | Hacker News Reader

Comment by trabant00 | Hacker News Reader

I think most here would know that math is not complete, consistent or decidable. (https://www.youtube.com/watch?v=HeQX2HjkcNo) But I'm going to leave that aside as it's pretty high level math for me and I never run into those problems in my life.

My personal problem with math that prevents me from seeing it as "discovery of fundamental principles claimed to exist in an objective reality" is natural numbers. It's impossible for me to clearly find the number 1 (for example) anywhere. The boundaries between one unit and 0.X or 1.Y seem arbitrary and chosen to help us create models. Religion and philosophy deal with this, but it's also apparent in digital signal processing. Or medicine: are the numerous bacteria in our bodies us, or not? Is the heat radiation from my body me? What about the fact that 1 + 1 rarely if ever equals 2? Meaning that two things together physically interact to create more than the sum of both parts. From celestial bodies to a replicated database the complexity goes through the roof when you have more than one thing.

That's not quite correct. Systems of mathematics cannot be both complete and consistent, but incomplete systems of mathematics can be consistent. For example Presburger arithmetic is provably consistent. There are limits to consistency for sure, but that doesn't mean there's no such thing as mathematical consistency.

I'm not sure what you mean. Presburger arithmetic is famously complete. What a system can't be is consistent, complete, and strong enough to perform a Godel encoding (which requires something multiplication-like). Drop any of the three requirements and it's possible.

Inconsistent: trivial, from falsehood follows anything.

Incomplete: Peano.

Weak: Presburger.

The comment could be interpreted as meaning that such systems cannot be complete or consistent, I'm just pointing out they can be one or the other. As I understand it, it is possible to consistently prove and decide things in mathematics, just not everything. Godel proved limits to mathematics, not that mathematics doesn't work. That's all.

> Godel proved limits to mathematics, not that mathematics doesn't work

Nobody claimed it doesn't work. It clearly does. The question is if it's a fundamental property of the universe or just an useful but flowed human mental model.

Sure, personally I see it as a language for expressing relationships and processes. I expounded on this in detail in another comment.

> Inconsistent: trivial, from falsehood follows anything.

Only trivial if you accept the principle of explosion (ex falso quodlibet or ex contradictione quodlibet). If you reject it, you end up with paraconsistent logic, from which one can develop nontrivial inconsistent mathematics see https://plato.stanford.edu/entries/mathematics-inconsistent/ and https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/1263...

> Systems of mathematics cannot be both complete and consistent

No. They can't be at the same times complete, consistent, decidable and powerful enough to express arithmetic. You can do complete, consistent and decidable though.

>I think most here would know that math is not complete, consistent or decidable.

There is zero evidence ZFC is inconsistent.

Even if "1" does not exist in reality mathematics still describes fundamental universal principles. As long as you believe that these fundamental principles exist at all they exist as mathematical ones.

Not even hardcore Platonists would claim that the number 1 exists in physical reality. But that does not mean it doesn't exist in some abtract sense. You can construct Models of reality using the natural numbers and these models about real objects are just imperfect descriptions of reality.

There's also zero conclusive evidence that ZFC is consistent. And even worse: if you found a proof (within ZFC or a weaker system) that ZFC was consistent, you would immediately know (by Gödel's second theorem) that it is actually inconsistent. The most we could hope for is that we couls prove its consistency in another system (one that hopefully convinces us more of its evident truth?).

ZFC is weird (especially choice). It's not implausible, but there's little a priori reason to assume that it describes some phyiscal reality. It just happens to give a foundation to a lot of really useful mathematics.

You could take a theory such as Peano Arithmetic and argue that that one is self-evident. But unfortunately, again by the second theorem, you can't use PA to prove ZFC consistent. That's, roughly, what Hilbert wanted to do in order to convince his critics, and he failed.

Human thought is paraconsistent - relevance/relevant logic best models how implication works in natural language, and that is paraconsistent; I think the intelligibility of inconsistent fiction such as Graham Priest’s Sylvan’s Box [0] is also evidence of that. If one believes mathematics is ultimately grounded in human thought, and if human thought is ultimately paraconsistent, that suggests paraconsistent logic may be a better foundation for mathematics than classical logic. It also suggests that maybe we should seriously consider taking the inconsistency horn of Godel’s trilemma (incomplete or inconsistent or weak), given the paraconsistent rejection of the principle of explosion means that doing so is non-trivial. Inconsistent theories can be strong, complete and non-trivial.

[0] https://projecteuclid.org/journals/notre-dame-journal-of-for...

I would bet that, even if ZFC is not consistent, there's another set of axioms which is, and in which all the stuff we've proven in ZFC still holds. That is, ZFC just happens to be a useful framework for the mathematics we're interested in. Even if ZFC collapses it seems very unlikely that all the stuff we've proven within it will; instead, we'll fix ZFC, like ZFC "fixed" naive set theory.

"1" works just fine for things that are discrete. Take eggs, for example. I have one egg. If it's smaller than other eggs, I don't have 0.9 eggs; no, I have exactly one egg or, if you prefer, I have exactly one small egg, but still exactly one.

I have exactly one wife. If she gained weight, I would not then have 1.01 wives.

I have exactly one cat. If she had N kittens, I would then have exactly N+1 cats, not (1 * N/10) cats.

And so on.

My looking at natural numbers gave me the exact opposite conclusion.

Once the grey blob of infinite nothingness becomes distinct enough that there is a difference between one moment/point and another moment/point, you have enough to start counting different states (or do binary arithmetic). Its high school math to find out some numbers are different then others and eventually you'll find the primes.

I'll bet there is no god, power, or alternative rules in any possible universe, fictional or real, that could not find the primes.

> It's impossible for me to clearly find the number 1 (for example) anywhere.

what about that there is one of you?

Reply on news.ycombinator.com