Theorem vs. Paradox

 In case you're in the mood to twist your mind a bit into knots, there's this:  ;)

https://inference-review.com/letter/a-theorem-and-a-paradox

Michael Harris on Mathematics by Machine

 Peering at the future of mathematics... and the philosophy underlying it?:

https://siliconreckoner.substack.com/p/can-mathematics-be-done-by-machine

Philosophizing With J. D. Hawkins

 IF you can find the time for it, a 2+ hour new interview (podcast) with mathematician/philosopher Joel David Hamkins:

https://www.youtube.com/watch?v=acjJ5-OSuZM

I Think Therefore I Am....

 Ambling a bit further from mathematics, an extensive listing of philosophy podcasts:

https://truesciphi.org/phipod_series.html

Must Human Civilization Have Numbers...

 Lengthy, interesting, thought-provoking new piece from Stephen Wolfram asking 'if numbers are inevitable?' (and concluding that at least for now, for humans, they are):

https://writings.stephenwolfram.com/2021/05/how-inevitable-is-the-concept-of-numbers/

Harvey Friedman and More

Very interesting piece in Nautilus this morning on Harvey Friedman and foundational aspects of mathematics:

http://nautil.us/issue/45/power/this-man-is-about-to-blow-up-mathematics

Dr. Friedman has a YouTube website here:
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ

We All Love Lists, Right...

Not much math here, but I'm intrigued a bit by philosophy of science (as an outside observer), so perhaps some of you are as well. A h/t to physicist Sean Carroll for recently pointing out this compiled list of "best Anglophone philosophers of science since 1945":

http://civs.cs.cornell.edu/cgi-bin/results.pl?id=E_81c68ea1c38fe926

Admittedly, I'm not even familiar with 2/3 of the 100+ names on this list (including "Bas van Fraassen," #5 in the rankings), but was still surprised by a few things about it:

1)  I enjoy reading Deborah Mayo's "Error Statistics Philosophy" blog, and so was unexpectedly pleased to find her ranked 59th out of 104 on the list.

2)  Despite the "popular" appeal of Thomas Kuhn's writings, surprised to see him at #2 in this list (thought his aura had faded somewhat, but apparently not).

3)  Also, didn't realize that Carl Hempel was viewed this highly -- could have anticipated a top 10 showing, but #3! and a notch ahead of Karl Popper at #4 would not have guessed. Feyerabend also higher than I would've expected (at #9).

4)  Surprised too that David Bohm even appears on this list (at #53), and the ubiquitous Wittgenstein does not appear at all (I assume his death in 1951 precluded inclusion here?).
But biggest surprise of all was Rudolf Carnap winning the poll fairly handily. He too (and "logical positivism," more generally) I thought had well-declined.  Apparently Martin Gardner was correct in predicting, years ago, a resurgence for Carnap (one of his professors), given some time! (a short piece by Gardner on Carnap is Chapter 3 in his great volume, "Are Universes Thicker Than Blackberries" HERE.)

Anyway, interesting compendium... any surprises for other readers?

Don't Mess With Popper ;-)

 Natalie Wolchover, ran a piece in Quanta recently with a title I love, "A Fight For the Soul of Science," covering some of the dissing of Popper falsification, in favor of more shoddy (IMO) induction-focused approaches (turning parts of modern-day physics into glorified metaphysics, by some accounts), leading to "a crisis" in which "the wildly speculative nature of modern physics theories... reflects a dangerous departure from the scientific method":

https://www.quantamagazine.org/20151216-physicists-and-philosophers-debate-the-boundaries-of-science/

As the article notes, "Theory has detached itself from experiment. The objects of theoretical speculation are now too far away, too small, too energetic or too far in the past to reach or rule out with our earthly instruments." That's a nice excuse for the science playground that has resulted, but in some form it could probably have been said at any point in the history of scientific method.
The discussion leads into Bayesianism (and specifically, "Bayesian confirmation theory"), and as always, Wolchover does a great job attempting to present different sides of a sticky topic. And I have no problem with (indeed I enjoy) speculative theorizing... I'm just unwilling to label it 'good science' (at best, it is good speculation, and that's often different).

Anyway, Andrew Gelman balanced some of the discussion with a more nuanced assessment, including lots of comments (and the debate goes on elsewhere, as well; see also an earlier Deborah Mayo take on Popperianism HERE):

http://andrewgelman.com/2015/12/17/gathering-of-philosophers-and-physicists-unaware-of-modern-reconciliation-of-bayes-and-popper/

In actuality, "the soul of science" has ALWAYS been threatened by different philosophical outlooks, but it ought be understood by all, that in general, "induction" (while necessary because it is unavoidable) is always a WEAK mode of empiricism, and it's no wonder a lot of folks are losing patience with the loosey-gooseyness in some areas of theoretical physics; a looseness that has long been present in biomedicine, psychology, economics, and some other areas, and in a kind of mission-creep (driven perhaps by academic/publication/career pressures), is now, to our detriment, expanding outward.

Laboring Over Formalism

Happy Labor Day to US readers... and here's a little reading you can labor over:

A piece (not sure how old it is???) from non-Platonist Timothy Gowers on philosophy and mathematics, focusing on Platonism, logicism and formalism (good stuff, BUT ONLY if you're already inclined toward philosophical underpinnings):

https://www.dpmms.cam.ac.uk/~wtg10/philosophy.html

It's brimming with interesting ideas, including (in the "#6 Truth and Provability" section) the notion that somewhere in the decimal expansion of pi there ought surely be a string of a million 7's, on the basis of it being a "normal number."

Here's a little bit of his wrap-up to the longread:

"...the point remains that if A is a mathematician who believes that mathematical objects exist in a Platonic sense, his outward behaviour will be no different from that of his colleague B who believes that they are fictitious entities, and hers in turn will be just like that of C who believes that the very question of whether they exist is meaningless...
"So why should a mathematician bother to think about philosophy? Here I would like to advance a rather cheeky thesis: that modern mathematicians are formalists, even if they profess otherwise, and that it is good that they are...

"When mathematicians discuss unsolved problems, what they are doing is not so much trying to uncover the truth as trying to find proofs….

"I also believe that the formalist way of looking at mathematics has beneficial pedagogical consequences. If you are too much of a Platonist or logicist, you may well be tempted by the idea that an ordered pair is really a funny kind of set -- the idea I criticized earlier. And if you teach that to undergraduates, you will confuse them unnecessarily. The same goes for many artificial definitions."

Math As Science

Waxing a bit philosophical today....

The debate over whether or not mathematics is a science goes back a long while. I won't address that argument specifically (since I think the underlying question of what is "a science" is unresolved and unresolvable -- the author below believes there exist "standard" operating definitions for science that work -- I don't believe those definitions can be used consistently, precisely, or unambiguously (and that has nothing to do with 'post-modernism' per se, as the writer would imply, but simply with the imprecise nature of language and limits/uncertainties of human cognition -- some biology IS science, some ISN'T; same for physics, geology, psychology, medicine, etc.).

But with that said, I did enjoy the approach and content of this 2008 piece on the subject, which touches upon several topics:

http://www.arachnoid.com/is_math_a_science/

In the end the author concludes:

"Nature is innately mathematical, and she speaks to us in mathematics. We only have to listen.
"Because nature is mathematical, any science that intends to describe nature is completely dependent on mathematics. It is impossible to overemphasize this point, and it is why Carl Friedrich Gauss called mathematics 'the queen of the sciences'."

 I have no qualms calling math 'the queen of the sciences' -- am just not sure that that means anything more than that math is the most logically precise/rigorous and least ambiguous of the large, continuous (not discrete) gradient of human thought processes.

The World As We Think Maybe Perhaps Possibly We Know It

Two great science communicators….:

I mini-reviewed Jim Holt's book "Why Does the World Exist?" awhile back at MathTango HERE, and he and physicist Sean Carroll recently sat down for an interesting discussion of the topic below:


Jim Holt and Sean Carroll from ALOUDla on Vimeo.


This is more physics and philosophy than mathematics, but still of potential interest to many math buffs. The conversation is close to an hour long, followed by audience questions, so you'll need to make some time for it, or play it in the background as you do other Webby things.

Unfortunately, they weren't able to settle the question once-and-for-all in the allotted time... ;-)

The Practical... and the Philosophical

1) Guillermo Bautista recently posted a list of 13 online sites giving calculus tutorials... might be useful for some (I'm not necessarily endorsing them, but just passing along, and also adding the link to the right-hand column "math instruction" list):

http://tinyurl.com/ajjy54d

2) A different blogger has interestingly spotlighted an older post by Terry Tao on 'rigor in mathematics' and intuitions:

http://m-phi.blogspot.com/2013/03/terry-tao-on-rigor-in-mathematics.html

May be a bit too philosophical for some, but anything from Dr. Tao of course is worth consideration.

Tao divides mathematical education into three stages: the "pre-rigorous," "rigorous," and "post-rigorous" stages. I like the approach he takes:

"The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient). So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. One way to do this is to ask yourself dumb questions; another is to relearn your field."

Read the entire post here:

http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/

Mystical Path... Mystical Math?

Popularizer Clifford Pickover often writes about the mystery and even mysticism of numbers. Paul Erdos was famous for saying certain (beautiful) mathematical proofs must come from 'God's book.' Lover of numbers, Martin Gardner. regarded himself as a "Mysterian" (and also a theist/fideist) who believed, despite the reality of numbers, humans could never fully comprehend the workings of their own minds. Cantor was deeply religious, writing proofs for the existence of God, which never gained the traction his proofs involving infinity did.

In short, I've always found fascinating the link many sense between math or numbers, and the mystical or Godly realm of existence. Math is often perceived, more than any other science, to somehow be associated with a deeper reality than we can otherwise be in touch with directly.
And yet, a different school of math, views math as little more than a creation or construct of the human mind; not so much existing in the 'world out there' so much as constrained to the world inside our heads.
Such basic, fundamental notions, yet leading to such divergent, unresolved thoughts.

Here's an old Julie Rehmeyer posting that touches on the subject (in which she quotes British mathematician Brian Davies as saying that Platonism “has more in common with mystical religions than with modern science"):

http://tinyurl.com/ycsn2bl

And lastly, if you have the time, Ben Vitale recently put up this hour+ long YouTube roundtable video on "Mathematics and Religion":


http://www.quora.com/Benjamin-Vitale/Mathematics-and-Religion

A Video Rerun

I've shown this before, but have several new readers now, and consider it worth repeating... An interesting 9-minute clip from an old "BloggingheadsTV" edition with John Horgan and Jim Holt discussing the nature of mathematics ('are mathematical truths discovered or invented?'), the Riemann Hypothesis, and similar topics:

Of Kuhn and Pythagorus

Errol Morris at the NY Times has been running a (5-part) series centered around philosopher Thomas Kuhn (of "paradigm-shift" fame). Part 3, is just out, and if historical (or, in this case, ancient) mathematics interests you, you may find his discussion of the Pythagoreans, "incommensurability," and irrationality of √2, quite fascinating --- the piece is long (and the extensive footnotes also worth reading), so only link to it when you have some time to sit and read:

http://opinionator.blogs.nytimes.com/2011/03/08/the-ashtray-hippasus-of-metapontum-part-3/

The first two parts of this engaging series are below:

http://opinionator.blogs.nytimes.com/2011/03/06/the-ashtray-the-ultimatum-part-1/

http://opinionator.blogs.nytimes.com/2011/03/07/the-ashtray-shifting-paradigms-part-2/

(For those of you who are Thomas Kuhn fans, I'll warn you that Morris is not.)

"What Is Mathematics?"

An ongoing debate:

http://members.cox.net/mathmistakes/what_is_mathematics1.htm

A discussion (actually review of a John Barrow book) centering around the long-time debate over whether mathematics objectively exists in reality or is merely a creation of the human mind.

...And plenty more food-for-thought here:

Nice article at Wikipedia giving overview of "philosophy of mathematics;" includes many links:

http://en.wikipedia.org/wiki/Philosophy_of_mathematics

...and similar "Foundations of Mathematics" article also at Wikipedia:

http://en.wikipedia.org/wiki/Foundations_of_mathematics

Relatedly, possibly worth mentioning a 2009 book from Jeremy Gray "Plato's Ghost: The Modernist Transformation of Mathematics," which narrates the evolution of mathematical thought from 1880 to the 1920s:

an online review here:

http://www.americanscientist.org/bookshelf/pub/modernism-in-mathematics