Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Doing Mathematics Differently

inference-review.com

Post reply on HN
Print view View original

51–60 of 73 posts

Re: Doing Mathematics Differently

#51

Post by mafribe »

Ono-Sendai wrote: post #18
mafribe wrote: post #17

Earlier quoted context omitted.

I'm sorry to say this, but your second post also doesn't really address the issue and continues to misunderstand the purpose and definitions of Kolgomorov complexity. The fact that for any specific string S you can find a language L such that the Kolgomorov complexity K(L, S) is 0, is not that interesting. 1. For any specific L, you can only play this trick (hardcoding a target string) for a finite number of strings.…

Regarding point 2, I agree, but the question is how to measure the increase in complexity - we are back at square one :)

You measure it in the same way, using the idea of shortest decription w.r.t. a fixed universal language. You than show that this is absolute up to a constant. Any results produced that way such as the incompressibility of most strings is independent of such constants, hence the the choice of universal language doesn't matter.

Re: Doing Mathematics Differently

#52

Post by plutooo »

tel wrote: post #37
plutooo wrote: post #24

Earlier quoted context omitted.

If you have different rules, you have a different object and the meaning of the axiom is different. Even if it is written using the same symbols. To continue the compiler analogy.. Just because the ASCII sequence "int c=0;" means different things in C and Java, doesn't imply "int c=0;" is meaningless when specifically talking about only C.

Not necessarily. It's very possible for one system to embed inside the other. In this case you can often talk about exactly the same axiom/theorem/whatever within different inferential systems. An interesting example is the Axiom of Choice which is, naively encoded, a theorem of intuitionistic logic. That said, we can use flattening to embed classical logic inside of intuitionistic logic and then recover a whole fami…

Still, "axiom of choice is a theorem in intuitionistic logic" is absolutely true whether or not you accept intuitionistic logic.

Re: Doing Mathematics Differently

#53

Post by plutooo »

mafribe wrote: post #50
catnaroek wrote: post #23

Earlier quoted context omitted.

Axioms and rules of inference are fundamentally different: (0) An axiom is an internal statement to a mathematical theory that is assumed to be true. That is, inside of a mathematical theory, you don't need to prove that its axioms hold. However, if you want to construct a model of a mathematical theory, you need to prove externally that the axioms hold. In return, you get the theory's theorems (suitably interpreted)…

Axioms and rules of inference are fundamentally different: Things are not that simple, because you can often convert axioms to rules of inference or vice versa, without changing the set of derivable consequences. As an example, consider pure first-order logic (FOL). As one extreme, you can present FOL with just one rule of inference (Modus Ponens), see for example [1]. The other extreme is Gentzen's sequent calculus…

Exactly. What are axioms and what are rules of inference ends up being a distinction without a difference.

Re: Doing Mathematics Differently

#54

Post by mafribe »

plutooo wrote: post #53
mafribe wrote: post #50

Earlier quoted context omitted.

Axioms and rules of inference are fundamentally different: Things are not that simple, because you can often convert axioms to rules of inference or vice versa, without changing the set of derivable consequences. As an example, consider pure first-order logic (FOL). As one extreme, you can present FOL with just one rule of inference (Modus Ponens), see for example [1]. The other extreme is Gentzen's sequent calculus…

Exactly. What are axioms and what are rules of inference ends up being a distinction without a difference.

I'm not sure I would go that far.

There is a difference, but it's not clear quite what it is.

For example, it becomes progressively harder to get nice (with cut elimination and finite rule schemata) sequent calculi for richer axiomatic systems. For example I have not come across a nice (in the above sense) sequent style formalisation of ZFC set theory. Have you?

Re: Doing Mathematics Differently

#55

Post by plutooo »

mafribe wrote: post #54
plutooo wrote: post #53

Earlier quoted context omitted.

Exactly. What are axioms and what are rules of inference ends up being a distinction without a difference.

I'm not sure I would go that far. There is a difference, but it's not clear quite what it is. For example, it becomes progressively harder to get nice (with cut elimination and finite rule schemata) sequent calculi for richer axiomatic systems. For example I have not come across a nice (in the above sense) sequent style formalisation of ZFC set theory. Have you?

No, but why is that a problem?

Re: Doing Mathematics Differently

#56

Post by mafribe »

plutooo wrote: post #55
mafribe wrote: post #54

Earlier quoted context omitted.

I'm not sure I would go that far. There is a difference, but it's not clear quite what it is. For example, it becomes progressively harder to get nice (with cut elimination and finite rule schemata) sequent calculi for richer axiomatic systems. For example I have not come across a nice (in the above sense) sequent style formalisation of ZFC set theory. Have you?

No, but why is that a problem?

It's an indication, a hint, that rules and axioms are not always completely exchangable.

Re: Doing Mathematics Differently

#57

Post by SeanDav »

plutooo wrote: post #19
smilliken wrote: post #9

This is a good article, and an important topic, but it mischaracterizes the discipline of mathematics: > the principle that mathematical truth is black or white and provides absolute certainty. > Pure mathematicians like to think that they have absolute truth Formal mathematics has no concept of absolute truth; this is left for philosophy. It's just concerned with axioms and theorems (and their proofs). Whether an ax…

"Formal mathematics has no concept of absolute truth" This is false. The axioms don't have to be true. You can still talk about their implications in absolute terms. "Assuming a=0 implies a=0" is absolutely true. Regardless of whether a actually is 0.

> "Assuming a=0 implies a=0" is absolutely true.

This is not correct. There is no such thing as absolute truth. Something can only be true within a set of previously agreed constraints and rules. For example, I can simply imagine a scenario where a=0 implies a=0 is considered to be false, because I define it to be so.

Re: Doing Mathematics Differently

#58

Post by __mbm__ »

bachback wrote: post #47
smilliken wrote: post #9

This is a good article, and an important topic, but it mischaracterizes the discipline of mathematics: > the principle that mathematical truth is black or white and provides absolute certainty. > Pure mathematicians like to think that they have absolute truth Formal mathematics has no concept of absolute truth; this is left for philosophy. It's just concerned with axioms and theorems (and their proofs). Whether an ax…

Have you followed Gödel's proof in detail? I know Fields medalists who like to speculate about Gödel, but haven't read Russell and Whiteheads Principia Mathematica (PM). Gödel's system is PM, but PM is quite an impossible read. Most of this discussed has been discussed over 30 years between 1900-1930. By the way Turing also builds on PM in his on computable numbers paper. PM is like a compiler before there were compi…

This is the crankiest comment I've seen on HN in a while. You "know Fields medalists"? Right.

You think that Pricipia Mathematica is the source of all truth in mathematics? Wrong. Most mathematicians working on foundational questions start out by learning Zermelo-Fraenkel set theory, which is well-understood, and avoids several difficulties that Russel had. While few people have read PM, it remains important because it goes through the hard task showing that high-level mathematics (calculus, &c.) can have rigorous, first-principle proofs.

"PM is like a compiler before there were compilers." What?

"I asked a math professor about the '=' sign and how things can be equal at all (...)" OMGWTFNO. I've seen comments like this before. Typically, it creates debate around a non-issue by sowing confusion and never nailing down precisely what we are talking about (hence the need for formal methods—it helps us call bullshit on comments like this).

"Two things are equal iff their transformation leads to the origin. [A - B = 0]". Wrong. There are equality relations that do not require the existence of a zero or the concept of addition or subtraction.

Re: Doing Mathematics Differently

#59

Post by pfortuny »

plutooo wrote: post #52
tel wrote: post #37

Earlier quoted context omitted.

Not necessarily. It's very possible for one system to embed inside the other. In this case you can often talk about exactly the same axiom/theorem/whatever within different inferential systems. An interesting example is the Axiom of Choice which is, naively encoded, a theorem of intuitionistic logic. That said, we can use flattening to embed classical logic inside of intuitionistic logic and then recover a whole fami…

Still, "axiom of choice is a theorem in intuitionistic logic" is absolutely true whether or not you accept intuitionistic logic.

But thas is not Mathematics, they call it "metamathematics" for something.

Re: Doing Mathematics Differently

#60

Post by meshr »

plutooo wrote: post #19
smilliken wrote: post #9

This is a good article, and an important topic, but it mischaracterizes the discipline of mathematics: > the principle that mathematical truth is black or white and provides absolute certainty. > Pure mathematicians like to think that they have absolute truth Formal mathematics has no concept of absolute truth; this is left for philosophy. It's just concerned with axioms and theorems (and their proofs). Whether an ax…

"Formal mathematics has no concept of absolute truth" This is false. The axioms don't have to be true. You can still talk about their implications in absolute terms. "Assuming a=0 implies a=0" is absolutely true. Regardless of whether a actually is 0.

As it was mentioned earlier we can’t think about “absolute truth" in terms of math because math is appliance science. It is like a language. The root science is physics and only it owns “absolute truth", the real axioms and implication rules.
Post reply on HN