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The Denver Philosophy Meetup Group Message Board › Godel, logicism, and mathematics

Godel, logicism, and mathematics

A former member
Post #: 272
Brian Z. said:

Ken, I agree that this will be an exciting conversation! You points are well taken, and I agree that Godel is probably too widely applied to other fields that don't have the systematic requirements that are needed to create something like the Godel sentence. Logicism is not a closed chapter in mathematics though. One way (but not the only way of course!) to asses an interpretation of math might be to see how mathematicians think about mathematics.

Right. This was the idea that Mill (among others) had. Husserl called this "psychologism" -- and made the concept implausible, I think. But Blanshard really hammered it to pieces.


From my experience in undergraduate and graduate mathematics, if a topic is covered with any rigor, it is covered axiomatically (which reeks of logicism). I'm not exactly sure how one could know that a theorem is true necessarily (what mathematicians are after) without the axioms and first order logic operating implicitly. Applied mathematicians and physicists might talk of math without reference to axioms or FOL, but if they rely at all on a theorem's truth, they are, I think, relying on axioms and FOL.

Yes, I agree with this, Brian. Very perceptive. And the interesting question is: what sort of a thing is an axiom, really?

Brian Z.
user 3263813
Denver, CO
Post #: 4
Ken, I just saw this! We missed you at the math group a few weeks ago. Perhaps a discussion of axioms at a future meeting would be helpful to all of us.
A former member
Post #: 292
Ken, I just saw this! We missed you at the math group a few weeks ago. Perhaps a discussion of axioms at a future meeting would be helpful to all of us.

I missed me, too. smile

Just lost track of the dates. (After doing all the reading, too.)

Yes, a discussion of axioms would hold interest for me, anyway.

Best,
Ken
Michael
Gravityisatheory
Denver, CO
Post #: 82
Hi Ken. I've joined this meetup, so nice to see you here. An axiom is a self-evident truth. Common axioms are of action, existence, and reason. You have to act in order to argue, you have to assume your existence, you have to use reason, etc. So any argument against these will necessarily use them, becoming self-refuting. For logic and mathematics it's first principles you assume without proof to study the consequences following this.
Brian Z.
user 3263813
Denver, CO
Post #: 6
Hey Michael, thanks for joining!

In any mathematical system, strictly speaking, reason or logic *itself* is not an axiom; rather, a logical system, is, as you hint at, the "backbone" of the entire enterprise. First order logic (FOL) consists of a set of symbols that can be combined into well formed formulas and set of rules of inference (e.g., modus ponens). In mathematics, axioms are introduced in order to add *content* to the set of uninterpreted symbols of FOL. So, for example, axioms for Euclidean geometry seem to make statements about the nature of space; then, with these unproved statements, one uses FOL to deduce the consequences that follow.

An interesting question arises (and I think this is why Ken and are are interested in 'meeting up' about axioms) when we ask questions like: "Why do we take these axioms to be true, and not others?" (it's not always the case that they are "self-evident"--see the example in geometry below); "Do the axioms chosen prove *all* of the true statements?" (the answer to this question for many logical systems is "no", a la Godel); "Do the axioms prove any false statements (e.g., is the system inconsistent?)?"

A really interesting historical example arises in the history of geometry. If one accepts Euclid's first four axioms but replaces the 5th axiom (the parallel postulate), which hounded many mathematicians precisely because it was not entirely self evident, with something else, one gets entirely new fields of "non-euclidean" geometries that are perfectly consistent! Then, it is an empirical question to ask whether the new geometries actually represent physical space. This is pretty remarkable stuff, I think!

Also, as a side note, there are philosophers who do try to use reason, not to *refute* reason, but to show the *limits* of reason (and they do so pretty successfully, many think). A few examples would be Hume, Kant, and our beloved Heidegger (who we're reading him this month!).
A former member
Post #: 293
Hi, Michael. This will be a great Meetup for you, I think, so welcome!

Yep, Brian is right on the money.

There's also a deeper question behind these questions concerning axioms. If we need axioms to get mathematics underway in any very substantial way, then it would appear that mathematics can't be pure logic, as some have contended. But, if it isn't pure logic, then it would seem to be logic plus -- well, something else. But what, really, is that "something else"? And how do we know it when we see it?

Incidentally, as you may know, not only can we get internally consistent non-euclidean geometries by replacing the 5th axiom, some of these new geometries have actually proven to have application in the real world (which is downright odd, in a way, and even not a situation that's unique to non-euclidean geometries). By which I mean that it often happens that some abstruse and apparently purely theoretical branch of mathematics will turn out to actually apply to reality in surprising ways.

So this thing "mathematics" would often appear to have some strange and even unexpected congruence with reality . . . and it may also be that axioms are abstracted from reality and imported into mathematics. . . .

--K.
Dan
danlg
Group Organizer
Broomfield, CO
Post #: 1,403
An interesting take on the venture you guys have been discussing: The Lucas-Penrose Argument about Gödel’s Theorem
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