(July 2, 2019 at 9:12 am)A Toy Windmill Wrote:

(July 2, 2019 at 7:50 am)Jehanne Wrote: Not as much, no doubt; how many papers have been published in the peer-reviewed mathematical journals by finitists over the last 50 years?  How  many textbooks are there devoted to the subject?  Of the 3K+ 4-year colleges in North America, where can I, as a student, take a class on finitism?  Are there any journals devoted to finitism?

It's not clear to me what anxieties are soothed by finitism that are not adequately soothed by intuitionism, which agrees that mathematical constructions are finite and which enjoys far more active research. It also has practical advantages to computer scientists.

CS does not have any issues with Cantorian mathematics, such as infinite Turning machines.

(July 2, 2019 at 11:36 am)Jehanne Wrote:

(July 2, 2019 at 9:12 am)A Toy Windmill Wrote: It's not clear to me what anxieties are soothed by finitism that are not adequately soothed by intuitionism, which agrees that mathematical constructions are finite and which enjoys far more active research. It also has practical advantages to computer scientists.

CS does not have any issues with Cantorian mathematics, such as infinite Turning machines.

And intuitionistic mathematics doesn't have issue with Turing machines either. They aren't particularly "Cantorian."

The most notable attraction of intuitionism, however, is in type theory, where type systems display a glaring analogy with logic that is celebrated as the Curry-Howard Correspondence. This has been upgraded to The Holy Trinity with the ever growing popularity of category theory in CS. The analogy is nearly always specifically with an intuitionistic logic, being also a logic that is automatically focused on computable functions, which CS people have a bias towards.

Personally, I like a full-blown class theory with a class version of the axiom of choice.

I guess that makes me not a finitist......

(July 2, 2019 at 12:19 pm)A Toy Windmill Wrote:

(July 2, 2019 at 11:36 am)Jehanne Wrote: CS does not have any issues with Cantorian mathematics, such as infinite Turning machines.

And intuitionistic mathematics doesn't have issue with Turing machines either. They aren't particularly "Cantorian."

The most notable attraction of intuitionism, however, is in type theory, where type systems display a glaring analogy with logic that is celebrated as the Curry-Howard Correspondence. This has been upgraded to The Holy Trinity with the ever growing popularity of category theory in CS. The analogy is nearly always specifically with an intuitionistic logic, being also a logic that is automatically focused on computable functions, which CS people have a bias towards.

Then "intuitionistic mathematics" is not finitism.

(July 2, 2019 at 11:29 pm)Jehanne Wrote:

(July 2, 2019 at 12:19 pm)A Toy Windmill Wrote: And intuitionistic mathematics doesn't have issue with Turing machines either. They aren't particularly "Cantorian."

The most notable attraction of intuitionism, however, is in type theory, where type systems display a glaring analogy with logic that is celebrated as the Curry-Howard Correspondence. This has been upgraded to The Holy Trinity with the ever growing popularity of category theory in CS. The analogy is nearly always specifically with an intuitionistic logic, being also a logic that is automatically focused on computable functions, which CS people have a bias towards.

Then "intuitionistic mathematics" is not finitism.

I wrote:

Quote:It's not clear to me what anxieties are soothed by finitism that are not adequately soothed by intuitionism, which agrees that mathematical constructions are finite

I am not saying that intuitionistic mathematics is finitism (no need for quotes: these are the proper terms). Finitism does not permit universal quantification over infinite domains in anything other than
a schematic form. Intuitionistic mathematics does. Both, however, assume that mathematical constructions are finite, and I'm not sure what anxiety finitism alleviates that intuitionism does not.

(July 3, 2019 at 2:52 am)A Toy Windmill Wrote:

(July 2, 2019 at 11:29 pm)Jehanne Wrote: Then "intuitionistic mathematics" is not finitism.

I wrote:

Quote:It's not clear to me what anxieties are soothed by finitism that are not adequately soothed by intuitionism, which agrees that mathematical constructions are finite

I am not saying that intuitionistic mathematics is finitism (no need for quotes: these are the proper terms). Finitism does not permit universal quantification over infinite domains in anything other than
a schematic form. Intuitionistic mathematics does. Both, however, assume that mathematical constructions are finite, and I'm not sure what anxiety finitism alleviates that intuitionism does not.

What mathematical constructions are infinite?

(July 3, 2019 at 11:51 am)Jehanne Wrote:

(July 3, 2019 at 2:52 am)A Toy Windmill Wrote: I wrote:

I am not saying that intuitionistic mathematics is finitism (no need for quotes: these are the proper terms). Finitism does not permit universal quantification over infinite domains in anything other than
a schematic form. Intuitionistic mathematics does. Both, however, assume that mathematical constructions are finite, and I'm not sure what anxiety finitism alleviates that intuitionism does not.

What mathematical constructions are infinite?

All uncomputable reals, which is most of them according to classical mathematics. We might argue that such things are not "constructions", in which case, swap "mathematical object" for "mathematical construction" in my previous posts.

(July 3, 2019 at 1:14 pm)A Toy Windmill Wrote:

(July 3, 2019 at 11:51 am)Jehanne Wrote: What mathematical constructions are infinite?

All uncomputable reals, which is most of them according to classical mathematics. We might argue that such things are not "constructions", in which case, swap "mathematical object" for "mathematical construction" in my previous posts.

My point is that Cantor, using a finite number of symbols, proved that some infinite sets (the Reals) are infinite and uncountable:

Wikipedia -- Cantor's diagonal argument

The rational numbers are, however, countable:

Wikipedia -- Countable set

And, so, what's the issue here?  Is finitism any different than Creationism?

(July 3, 2019 at 10:06 pm)Jehanne Wrote:

(July 3, 2019 at 1:14 pm)A Toy Windmill Wrote: All uncomputable reals, which is most of them according to classical mathematics. We might argue that such things are not "constructions", in which case, swap "mathematical object" for "mathematical construction" in my previous posts.

My point is that Cantor, using a finite number of symbols, proved that some infinite sets (the Reals) are infinite and uncountable:

Wikipedia -- Cantor's diagonal argument

The rational numbers are, however, countable:

Wikipedia -- Countable set

And, so, what's the issue here?  Is finitism any different than Creationism?

Where is the proof that some sets are infinite in the first place? Cantor didn't prove the existence of such things Their existence is only granted axiomatically in set theory.

A finitist can accept Cantor's diagonal argument as a scheme to transform a finitistic function --- which the classical mathematician calls an enumeration of the reals --- into another function --- which the classical mathematician thinks gives a real not in the enumeration.

(July 4, 2019 at 3:42 am)A Toy Windmill Wrote:

(July 3, 2019 at 10:06 pm)Jehanne Wrote: My point is that Cantor, using a finite number of symbols, proved that some infinite sets (the Reals) are infinite and uncountable:

Wikipedia -- Cantor's diagonal argument

The rational numbers are, however, countable:

Wikipedia -- Countable set

And, so, what's the issue here?  Is finitism any different than Creationism?

Where is the proof that some sets are infinite in the first place? Cantor didn't prove the existence of such things Their existence is only granted axiomatically in set theory.

A finitist can accept Cantor's diagonal argument as a scheme to transform a finitistic function --- which the classical mathematician calls an enumeration of the reals --- into another function --- which the classical mathematician thinks gives a real not in the enumeration.

Do finistists accept the existence of irrational numbers? Just curious. I know that you may be playing the DA here.