[FOM] l basis of Intuitionism as in geometry

Adriano Palma Palma at ukzn.ac.za
Fri Jun 7 02:35:25 EDT 2013

Colin McLarty has the useful suggestion.
If you are interested in it I suggest to look at Giovanni Girolamo Saccheri (who precisely communicated his skepticism about axiomatic -- Euclidean style, we are way before Hilbert and the Goettingen school- geometry.)
The point I am making is that he did without giving the axioms of say a Riemannian or Lobatchevskyii space.
Equally of interest, while nobody -- to my knowledge- is aware of Saccheri knowing the source,. in the works by Khaya'ham you find the discussions of the difficulties in Euclid.
I came to know that one member of the logical brotherhood (dr pr. W Hodges) is really working on the arabic logical approaches to such problems.
I am not an expert in history but it is a good way to see how someone who has an utterly different approach w.r.t. an axiomatic system can go about looking at alternatives. What is interesting of the works is that they are independent from smething yo -or a contemporary- can find easy (from GRT, e.g.) i.e. conceive of curved timelines, spacetime which is a Riemannian manifold or some such, now familiar from physics.
With my best regards

From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Colin McLarty
Sent: 06 June 2013 06:11 AM
To: Foundations of Mathematics
Subject: Re: [FOM] Psychological basis of Intuitionism


I am not sure exactly what you mean by the question.  But could you answer your question by reading the 18th century mathematicians who actually did it?

best, Colin

On Wed, Jun 5, 2013 at 10:42 AM, Steve Stevenson <steve at clemson.edu<mailto:steve at clemson.edu>> wrote:
On Tue, Jun 4, 2013 at 9:41 AM, Andrej Bauer <andrej.bauer at andrej.com<mailto:andrej.bauer at andrej.com>> wrote:
>If mathematics is the art of hypothetical reasoning, surely then we must
> develop mathematics which is open to all possibilities, including the one in
> which mathematics has an independent and objective nature, and moreover
> allows for existence without construction.

I am asking a different question. Suppose you're an 18th Century
geometer and want to develop a non-Euclidean geometry. Since there are
no known such, how do you think about reformulating the parallel line
axiom? Once you've put a theory together, how do you communicate this
insight without show your colleagues how to construct such a geometry?

D. E. (Steve) Stevenson, PhD
Emeritus Associate Professor
School of Computing, Clemson University

"Those that know, do. Those that understand, teach," Aristotle.
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