Fwd: [cafephilodcdialogue] 7/25/10 Café Philo DC: "Is Mathematics Merely a Human Construct?”
Paul Gully
here are the aforementioned details, plus a shiny new topic to hold discussion in! woot.
SSPC: RISE!
From: kenfphilo <kenf...@aol.com>
Date: Tue, Jul 20, 2010 at 9:10 AM
Subject: [cafephilodcdialogue] 7/25/10 Café Philo DC: "Is Mathematics Merely a Human Construct?”
To: cafephilo...@yahoogroups.com
Greetings:
The next meeting of Café Philo DC will take place this coming Sunday, July 25, 2010 from 1 PM to 3PM at Reiter's Book Store, 1900 G St., NW, Washington, DC. The topic will be: "Is Mathematics Merely a Human Construct?". Avinash Patwardhan will moderate the session.
As background for the discussion, I offer below the comments of this topic's proposer, Hewitt Rose, who explains his understanding of the question and provides some well-researched suggested readings.
Hewitt's take on the topic:
In any rigorous epistemological argument, eventually someone is going to rear up and say, "Well, there is one example of Absolute Truth -- something that we can be certain is true for everyone, everywhere, at all times -- mathematics. The claim 1 + 2 = 3 is Absolutely True; even God cannot get a different result." Someone else will reply, "The claim is only true because we define 1, +, 2, =, and 3 that way and because of the context of the claim. If we toss one glass of water into the ocean, then two more, do we have three glasses of water? We say, 'no' only because of the categories that we create." The argument is not merely over math, but rather the nature of truth. If, for example, you can demonstrate that there is absolute mathematical truth, does that not argue for the possibility of absolute moral truth or some meaning to life that is not merely subjective?
So, is mathematical truth out there to be discovered, is it simply one of many possible ways we can conceive of our world, or is it both? Math works in the real world astonishingly well, yet math is unlike any other human knowledge. There is no way to empirically test the ultimate validity of math itself. In addition, there is no way to build up math rationally from first principles. Alfred North Whitehead and Bertrand Russell tried in the Principia Mathematica, only to be forced into admitting they were wrong by Godel's Incompleteness Theorem. So much for mathematical certainty, and so much for any certainty. Still, if math is only a human construct, then it is certainly a construct that works with great reliability and consistency between people; that is, it appears to be objectively true. Gregory Chaitin suggests that if a mathematical claim performs reliably, i.e., it passes the test of experiment, then we should treat it as objectively and relatively true, even if we cannot prove it in a formal sense.
That point leads to the problem of computer proofs of mathematical statements. Some computer proofs are so long as to be incomprehensible to anyone. Can we accept them as proofs just because the computer says so, unverified by any human, or just because the claim seems reliable? If we do, are we treating the computer like a God; that is, are we taking the computer's authority on faith?
Neurologists and biologists have also entered the debate, pointing to structures in the brain that are hard-wired to do mathematics. Biologists have found that many other animals can make surprisingly sophisticated calculations. So, does evolution reveal mathematical truth, better selecting for those animals that add 1 + 2 and get 3 instead of 4? Does our brain structure limit our mathematical understanding in ways that do not burden computers? (Computers have come up with comprehensible, even simple, geometric proofs previously unknown to humanity.) Is this an example of proof or of experiment? Is there a legitimate difference?
Some possible readings:
Fernando Gouvea, Book Review of Reuben Hersh's What is Mathematics, Really? (Concise stage setting and discussion of Hersh's concept of math as a socio-cultural construct, not merely a subjective construct.)
http://www.maa.org/reviews/whatis.html
Robert Lawrence Kuhn, Is Mathematics Invented or Discovered? (Survey of highly informed opinions, stimulating)
http://www.scienceandreligiontoday.com/2010/04/01/is-mathematics-invented-or-discovered/
Godel's Incompleteness Theorem. (The mathematically light philosophical version)
http://www.miskatonic.org/godel.html
Steven Krantz, The Proof in the Pudding (334 page book, dip into sections, particularly computer proofs. Accessible, although long).
http://www.math.wustl.edu/~sk/books/proof.pdf
Gila Hanna, The Ongoing Value of Proof (nominally about math education, in fact an argument that top-down formal proof is essential for math.)
http://fcis.oise.utoronto.ca/~ghanna/pme96prf.html
Wikipedia, Philosophy of mathematics (20 page survey, but good.)
http://en.wikipedia.org/w/index.php?title=Philosophy_of_mathematics&printable=yes
Snowball Dances to Another One Bites the Dust (Music is math applied, and music appreciation is not unique to people, as the video proves.)
http://www.youtube.com/watch?v=cJOZp2ZftCw
Lastly, for those who plan to attend Sunday's meeting and participate in selecting the subsequent topic, here are some suggestions culled from previous runners-up vote getters:
o Is Freedom or Equality Preferable When They Conflict?
o Does the Validity of a Belief System Depend on Its Success in Practice?
o Is the Possibility of Death Necessary for Moral Action?
o If We Could Eliminate All Pain, Should We?
o What is the Difference Between Man and a Machine?
o Is Inaction a Choice?
o Does Honesty Matter?
o Is Personal Reinvention an Act of Insincerity?
Best regards,
Ken Feldman
Founder, Café Philo DC and
Founder/Moderator, Café Philo DC Dialogue
KenF...@aol.com
Michael Oppenheimer
Quoc-Thuy Vuong
Cheers, aru
Paul Gully
promise that. (nor does your coming preclude the possibility of
inception again, btw). we would be pleased if you would come, but dont
feel obliged
Thuy: yes! absolutly. live discussion may be better suited, but still.
(that invitation also goes out to you if your in town)
(as it does to anyone)