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

61–70 of 73 posts

Re: Doing Mathematics Differently

#61

Post by plutooo »

SeanDav wrote: post #57
plutooo wrote: post #19

Earlier quoted context omitted.

"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.

"There is no such thing as absolute truth."

Isn't that an absolute truth?

Re: Doing Mathematics Differently

#62

Post by jeffreyrogers »

gaur wrote: post #7

What's the point of inserting untranslated French text into an English article? Is it supposed to just be window-dressing? Is it supposed to promote the (outdated and highly dubious) notion that all educated people speak French? Is it just for the author to show off? Whatever the reason, it's a highly obnoxious practice and it doesn't improve the article.

I found that slightly annoying as well, since elsewhere on the site they translate from French.

For mathematicians of a certain age knowing how to read French, English, and German is pretty common since often mathematics papers were only available in one language. So, my guess is that because of when the author was educated this is just normal for him.

Re: Doing Mathematics Differently

#63

Post by aangjie »

dllthomas wrote: post #29
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)…

This seems to be a technical objection that dodges the meat of the parent's comment. In what sense is it true that a given theorem follows from a given set of axioms and a given choice of inference rules?

Umm. I've been of the understanding that we never change the inference rules, Can you point to some branch of mathematics that involves a change of inference rules?

Re: Doing Mathematics Differently

#64

Post by aangjie »

dllthomas wrote: post #29
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)…

This seems to be a technical objection that dodges the meat of the parent's comment. In what sense is it true that a given theorem follows from a given set of axioms and a given choice of inference rules?

Umm. I've been of the understanding that we never change the inference rules, Can you point to some branch of mathematics that involves a change of inference rules?

Re: Doing Mathematics Differently

#65

Post by eli_gottlieb »

plutooo wrote: post #15

Omega is an amazing number. We know it has a digit in its binary expansion that is 0 or 1, yet it is impossible to formulate why. It is impossible to come up with a train of thought that explains it. There is no explanation that can be written down on a piece of paper. So we are left with the weird conclusion: 1. It has the value it has for no reason. 2. It has the value it has for a reason that is impossible to form…

Nah, Omega is just defined as a number which contains countably infinite bits of algorithmic information. Since algorithmic information is equivalent to thermodynamic information (and likewise relates to quantum information if you head in that direction...), the whole "you can't calculate Omega" thing is really just a way of saying, "You can't have an infinite-precision physical measurement encoded into a finite physical system."

Chaitin makes far too much metaphysics of his work.

Re: Doing Mathematics Differently

#66

Post by thomasahle »

bachback wrote: post #48
plutooo wrote: post #19

Earlier quoted context omitted.

"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.

"The axioms don't have to be true." No, not at all. Axioms are better thought of as universally accepted truths which everything else depends on. https://en.wikipedia.org/wiki/Logical_atomism

"Axioms are better thought of as universally accepted truths" Not really universally accepted. You just have to specify which set of axioms you are using. Different problems may call for different axioms. Like if you want to prove theorems about the security of some computer system, you'll define a set of axioms that allow proving interesting results for that purpose. If you want to prove theorems about sets, you'll want to specify what set axioms you are using. You may even try to specify interesting minimal sets of axioms that make your theorem true. There is no reason to use the same axioms for everything, or to assume they are True in some teological sense.

Re: Doing Mathematics Differently

#67

Post by Houshalter »

zvrba wrote: post #44
gizi wrote: post #42

Earlier quoted context omitted.

> Formal mathematics has no concept of absolute truth ... Well, it does. If you drag the propositional calculus into the fray, "true" is arbitrarily but probably most elegantly defined as following: true(x) = x or not x False is defined like this: false(x) = x and not x These definitions are pretty much arbitrary. The lambda calculus does this: true(a,b)=a false(a,b)=b It also works absolutely fine. > It's just conce…

Well, it does not. Mathematics is a symbolic system invented by humans, but that system tells you nothing about the state of the world . Used correctly, it can be used to describe the world, but it can also be used to describe phenomena that have not been physically observed. E.g., currently accepted theories allow for existence of wormholes, but we do not know whether the actual physical laws of the universe ("absol…

Yes there are absolute truths. 2+2 will not equal 5 in any universe. The laws of physics could be radically different, but I have 2 apples, and you give me two more, I will not suddenly have 5 apples.

Math can tell you an awful lot about the state of the world. That's why it's useful.

Re: Doing Mathematics Differently

#68

Post by Houshalter »

plutooo wrote: post #15

Omega is an amazing number. We know it has a digit in its binary expansion that is 0 or 1, yet it is impossible to formulate why. It is impossible to come up with a train of thought that explains it. There is no explanation that can be written down on a piece of paper. So we are left with the weird conclusion: 1. It has the value it has for no reason. 2. It has the value it has for a reason that is impossible to form…

I don't really understand the insight with Chaitin's number. It's unknowable, but so are a lot of things. The halting problem already says that there are programs which we can't know if they will halt or not. And before that, Godel showed there are simple logical sentences which are true and unprovable.

Re: Doing Mathematics Differently

#69

Post by catnaroek »

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…

> 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.

Yes, but the derivations themselves will change.

> 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 [2] which has only one axiom (A |- A), everything else being a rule of inference. Most presentations of FOL are between these extremes.

Yep. I'm aware of the phenomenon that a single mathematical object of type T (say, infinity-categories) may admit multiple presentations by objects of type T' (say, model categories). But, just because two objects of type T' present the same object of type T (e.g., two Quillen-equivalent categories), it doesn't mean that they are equal in all respects (e.g., the category of simplicial sets is much nicer than the category of topological spaces).

Re: Doing Mathematics Differently

#70

Post by erichocean »

plutooo wrote: post #61
SeanDav wrote: post #57

Earlier quoted context omitted.

> "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.

"There is no such thing as absolute truth." Isn't that an absolute truth?

Although I know you meant that jokingly, presumably the op's statement is only true within the parameters of a world where "there is no such thing as absolute truth", therefore, the statement isn't absolutely true, it's just true within the confines of his own (subjective) world view. The actual world, may, in fact, have absolute truth.
Post reply on HN