Math is a really useful subject—at least, that's what your parents and teachers told you. But math also leads to scenarios, like Zeno's paradoxes, that seem to inspire skepticism. So why do we believe in math and rely on it to build bridges and spaceships? How can anyone discover the secrets of the universe by simply scribbling numbers on a piece of paper? Is math some kind of magic, or does it have a more ordinary explanation? And could math be culturally relative, or are its concepts timeless and universal? Josh and Ray add things up with Arezoo Islami from SF State University.
Is math a realm of timeless, universal truths? If equations are made up, how do we know they’re right? Josh questions why and how math is useful, and he thinks that math doesn’t tell us anything new about the world. In contrast, Ray believes that math gets at deep truths and describes the fundamental structure of the universe.
The hosts welcome Arezoo Islami, Professor of Philosophy at San Francisco State University, to the show. Arezoo discusses how math is an invention on the basis of discovery along with the relationship between geometric ideals and real-life measurements. Ray asks about the limitations of Euclidean geometry, and Arezoo explains how different systems of geometry are necessary to study different universes. Josh questions why imaginary numbers were so controversial when they were first discovered, and Arezoo describes the importance of new ideas in math fitting in with the rest of mathematics. She believes that math is a conceptual tool that helps us solve problems in the world and extends our reach in the universe.
In the last segment of the show, Josh, Ray, and Arezoo discuss the limits of mathematical knowledge, being naturally talented at math, and its ability to connect people across time and cultures. Arezoo believes that math helps us grow our knowledge of the universe, but Josh pushes back by saying that it might show us the limits to human knowledge instead. Ray asks why revolutions don’t exist in mathematics the same way they do in science, but Arezoo points out that they are simply unrecognized in the way we learn math in schools. Given the power, she would have more philosophers teach math and give everyone a chance to fall in love with its beauty regardless of their natural talent.
Josh Landy
Is math a realm of timeless universal truths?
Ray Briggs
Or are mathematicians just making it up as they go along?
Josh Landy
If equations are made up, why are they so useful?
Comments (58)
Harold G. Neuman
Wednesday, August 11, 2021 -- 8:22 AM
I think I sorta get the ideaI think I sorta get the idea that math is thought of as timeless. Still, that idea seems metaphysical in some sense. I mean, if we accept that even life itself is not timeless---geological science and the fossil record suggest this---how might we square such a notion with the origins of man? I like my brother's characterization of metaphysics as a 'wild-ass guess'..But then, is metaphysics also timeless? It would not seem so.
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Tim Smith
Friday, August 27, 2021 -- 10:18 AM
Plato tied the concept ofPlato tied the concept of forms to western thought like a gordian knot. That Einstein revisited Bernard Reiman's topology to mathematize gravity counterfactually reinforced the idea that math is an a priori act of discovery. It is not.
Mathematics and logic seem to suggest a priori knowledge without the need for experience. But that knowledge was never preexisting to the human act of investigation. The timelessness that imaginary numbers undermine is the product of these investigations.
That some math precedes its application is only a reflection of the gurgling intellectual history of its creation. Bernard Rieman pushed the Zeta function into the complex plane and thought of the curvature of space due to mathematical tools taken from space and motion. Rieman defined the integral, not Leibniz or Newton. Hilbert corrected Rieman's Dirichlet principle. Einstein took a decade to work out the math to explain general relativity.
Math attaches itself to the brain that creates it. It is enduring in the culture that teaches it. It is deceptive in the view that confuses truth with beauty. Repeatability is the only measure of progress worth rigor. The empirical law of epistemology makes sense. Arezoo Islami seems poised to sever this knot.
Viewing information algorithmically is a new mathematical tool. Islami's philosophy promises the tools to come with the proper philosophical foundation from which to make those investigations. If her work is understood, I would think much money and time could be saved thinking of the problems we face and not false visions of preexisting depth and meaning – like space travel and plastic islands floating in the middle of our oceans.
Godel proved some true theorems cannot be proven. Turing showed that some functions cannot be computed. Dr. Islami is suggesting math offers tools that can be, and necessarily are, constructed.
Islami's view is deep, and not widely accepted by practitioners of technology and science. I look forward to corrections if this is not her tack.
Tim Smith
Wednesday, October 13, 2021 -- 7:24 AM
Well, no Gordian knot slicingWell, no Gordian knot slicing here, but still a great back and forth.
I especially enjoyed Ed Frenkel’s metaphor of Mathematics as an archipelago. Frenkel is an excellent argument for tolerance if ever there was one. If PT could get him on the show to talk about anything – we would all be the better for it.
Note we tried to change Mathematic education in the “New Math” movement. I would have liked to have heard Islami talk to that approach and maybe to the critiques of Morris Kline.
More back and forth between guest, Josh, and Ray might help. I felt pushed by Czar duties, this or that question. More call and response would have helped. I think Islami had more to give, misspoke a few times, and needed push back.
I ordered Yackel’s book. If you can’t cut knots, you might as well make them.
Harold G. Neuman
Tuesday, October 5, 2021 -- 8:28 AM
Whether math's beginningsWhether math's beginnings were as humble as finger counting or the abacus, it remains in time, rather than 'timeless'. (Sorry, Professor Dennett). Timelessness, like the current emphasis on all things awesome, is metaphoric. Or, perhaps, metaphysical? Seems to me, anyway.
es30
Thursday, October 7, 2021 -- 9:16 PM
"So why do we believe in math"So why do we believe in math and rely on it to build bridges and spaceships?" Good question. I look forward to hearing whether Eugene Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" makes it into the conversation.
Tim Smith
Wednesday, October 13, 2021 -- 6:55 AM
es30,es30,
Arezoo doesn’t specifically address Wigner’s “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” in this discussion but the ideas run throughout the show.
Look to her review of the equally well-spoken Martin Carrier and Johannes Lehnard’s study “Mathematics as a tool: tracing new roles of mathematics in the sciences.” There she summarizes what I believe to be her take on Wigner.
“In an often quoted but not carefully read paper, Wigner (1960) spoke of the unreasonable effectiveness of mathematics in the natural sciences. Mathematics and modern physics in his view presented such fundamentally different branches of knowledge that their close relationship appeared to be puzzling, mysterious and unreasonable.
Missing from Wigner’s analysis is an account of the constitution of mathematics, on the one hand, and its changing relationship with sciences on the other. He adopts uncritically the definition of pure abstract mathematics of the twentieth century and projects it to the mathematics of all areas and all eras. Moreover, the paper is focused almost exclusively on modern theoretical physics (Islami 2016).
Thus, critics of Wigner have focused, and with good motivation, on ineffectiveness of mathematics in other sciences (see, for example, Longo and Monte ́vil 2013; Steiner 1998; Velupillai 2005, for responses).”
I wonder what you might think of that? Would the show have gone there, but it did not.
The recent clash between epidemiologists and virologists about the term “transmission” is an excellent example of where Wigner went wrong with that essay/talk. Arezoo gets into the weeds in her own work, saying Wigner is not to blame for the insult.
Regardless of who said what and when the idea that Math speaks to divinity was a mistake by Dr. Islami. I think she would regret using that phrase if your comment pressed her. Misinterpretation of Math as anything but a tool is a crucial talking point for Arezoo and a cluster for Wigner and most physical scientists today. The world will and has already started to pay the price for computational biologists philosophically getting this wrong.
Harold G. Neuman
Wednesday, October 20, 2021 -- 3:34 PM
I am of an opinion that math,I am of an opinion that math, as many other things, has been/(is?) emergent or evolutionary. Perhaps this is self-evident. As it became sophisticated and abstract, so did the thinking of man. As it said more of 'why not' rather than' why' [ding!], our horizons expanded and expectations were more easily realized: bridges were built; arches fashioned; atoms busted. Has mathematics reached its' limitations? If so what does this mean? If not, what is next? Moreover, maybe, is there another form of math, unknown to humans? That would change things, seems to me.
Harold G. Neuman
Thursday, October 21, 2021 -- 6:57 AM
Or, to illustrate my notionOr, to illustrate my notion of doing the best we can with what we have and know, the more we have and know about about math, the better we can do (usually). The controversy between the developers of calculus, Newton and Leibnitz(?), was more about pride of authorship than anything else. Great minds think alike. But, most want to be recognized as FIRST.
Harold G. Neuman
Sunday, October 24, 2021 -- 8:55 AM
May have a few more remarks,May have a few more remarks, if I can express them coherently. There's the rub...
Harold G. Neuman
Monday, October 25, 2021 -- 1:12 PM
The Mathematical Mystery ofThe Mathematical Mystery of Timelessness. Let's fathom that. In one dimension, infinity poses an enigma, useful in some conceptual sense. Or another. In an essay, I wrote of the functional impracticality of our notion of infinity, saying we cannot get there from here,because there is no there, there---and, even if there were, there would be no 'what' to realize. A pragmatist nightmare..
So to speak. This is fun. So is the metaphysic behind it. But, metaphysics is/are wild ass guesses, immeasurable in a meaningful sense. Math flirts with infinity, but the latter is an unwilling object. One does not capture the untouchable with anchors and chains. How's that for philosophy, comrades.?
Tim Smith
Wednesday, October 27, 2021 -- 10:07 AM
I like this post.I like this post.
Why math should ever be considered outside time is a question not asked that needs a response. Discussing infinity and time together is deceptively intuitive. Arezoo Islami is a philosopher of time as well as math. It would have been helpful to hear her talk to both projects and correlate anything.
Infinity is a tool and no enigma, however. It is practical and limiting, both. Limits are the chains, and rigor is the anchor that changed the conception of infinity as we considered both the infinite heaven and the infinitesimal. Math has done a pretty fine job of capturing infinity, for the most part, with the concept of limits. Hilbert's hotel presents a world with many infinities. The Grand Hotel is a world that is useful.
Wild ass guesses (WAGs) should be disambiguated from iWAGs – informed WAGs. Some of the most meaningful and deep anchors and chains of my life and thought, of any one person's thoughts, are WAGs. Some can be dispensed with thought and experience, however, as well as lurking in the unquestioned and pervasive ether of culture.
Yet another vector for caution. We need to be careful to question our wild assery before dismissing someone else's iWAG. The "I" in iWAG can be the source of that dismissal. But to term epistemology, as a WAG doesn't pull power from explanations; it only focuses on their importance.
Zeno's paradoxes are a lark, were dismissed in his lifetime, and have persisted to modern times in those who failed to come to terms with them philosophically. Democritus and Leucippus answered the paradoxes in real-time (in the lifetime of Socrates) but were themselves dismissed by Platonic and later Aristotelian intransigence. Leucippus, a possible student of Zeno's, proposed the atomic theory of matter in Sherlockian fashion. Unfortunately, Josh points to Zeno instead of Democritus, but people still get confused. It is hard to overcome Ancient thought, the Roman and Medieval and predominantly modern, still accepted, Aristotelian world views. Reacting, adjusting, and refuting them is incumbent on philosophers today. Not only to read Plato and Aristotle but to refute and adapt them. Zeno can be dismissed entirely, however, as can much of Parmenides (all Monism aside.)
Democritus and Lucretius, on the other hand, need appreciation. Lucretius has had some rejuvenation, but Democritus wasn't rethought until Einstein considered Brownian motion. This show would have been a great time to refocus on the philosophy of thought experiments, their consequences, and how math can model thought and, thereby, reality.
Ancient Greek life had the conflict of Sophistry (Fake News) and Reason (Math – "Let none ignorant of geometry pass"), but that conflict comes to us by the filtered quill of others who didn't understand or appreciate the thoughts and conflicts. These quills had their own ideas and conflicts, which also WAG the dog.
tebvas
Tuesday, November 28, 2023 -- 11:08 PM
Newton and Leibnitz (?) wereNewton and Leibnitz (?) were the two individuals who were responsible for the development of calculus. The disagreement between them was more about pride of authorship than anything else. Minds that are great think similarly. On the other hand, the majority of drift boss people desire to be acknowledged as FIRST.
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Harold G. Neuman
Sunday, January 28, 2024 -- 3:02 PM
I herewith retract most ofI herewith retract most of what I proffered to the philosophic universe, having recognized I was barking in the wrong forest. A lame apology follows:
* I now accept that math IS timeless., with one, or more, qualifier(s). An example is offered by avering that the rivalry between Newton and Leibniz, over Calculus occurred over a cosmicly miniscule increment of time.. The "discovery" was just awaiting a discoverer. Sooner, or later, someone would have figured it out. It need not have been either man. It might have been, for example, the Frenchman, Rene, accent ague . It was only a matter of time. Or, timelessness. Whew! That was tough for me.
* Couched differently,, we might claim math is a universal truth---to everything, there is a season, and so on. Is IT a universal truth in any other universe? I don't think we can know that it matters, or that we could know if it did..
* math, and its' sibling physics, help us feel at home in the universe...help us manipulate things to our advantage. Or not. Is the timelessness of math (or physics) mysterious? I don't think so now. It was only a matter of time.
Humbly,
Neuman.
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