Mathematics is also surprisingly useful within mathematics itself: once thought of as two unrelated disciplines, algebra and geometry are now routinely used for the mutual benefit.
The Utility of Mathematics
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Re: The Utility of Mathematics
#12This article reminds me of the story of Fourier. It's hard to tell the history of math without the way in which Fourier series and the heat equation shaped the development and increasing rigor in calculus. The mathematicians of the day were extremely skeptical of infinite sums of trigonometric functions because it didn't fit their "a-priori" model of calculus. >Here was the heart of the crisis. Infinite sums of trigo…
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> calculus was given a new name: Analysis
"Mathematical analysis of the infintesimal", peaked in the works of Euler, was just that: an attempt to mathematically analyze, among other things, the notion and applications of the infinitely small (and infinity in general); for some reason the words "mathematical" and "infinity" were subsequently dropped, and we are left just with a generic term "analysis" which now requires context to be understood properly.
Re: The Utility of Mathematics
#13[deleted]
Re: The Utility of Mathematics
#14I'm not knowledgeable enough about their works to say anything interesting, but on the philosophy side of this there were a lot of developments in the 20th century, notably the logical positivists (e.g. Carnap) and later Quine. Quine disagrees with the logical positivists in a way that I find a little tricky to pin down, despite his and their writing being much clearer than "continental" philosophers, but I have foun…
My mind went to Quine as well. One issue is that the empirical world is really some representation of it via human senses, or some consensual agreement on it. I'm not trying to question that there is an external reality, but (along the lines of Quine) one might argue that our understanding of it -- our perception of it, and mental representation of it -- is fundamentally a human representation. One might argue the sa…
Thank you. I've been circling this thought process for a while and didn't know if it is completely incoherent.
The thought process began thinking all mathematical equations can be translated to English and vice versa. There must be a shared structure to language and math, or an isomorphism at the least. The effectiveness of mathematics then is more about its conciseness, not that it says anything new about the world we don't have with language.
One "natural" idea is that both our internal representations and external reality depend on spatio-temporal relations. Through sptatio-temporal relations a sound might hit our left ear before our right, or that we have a memory of the past and not the future. Noticing these relations are what are brain is good at, because it too is spatio-temporally laid out. We could then imagine from this base set of relations we construct language and math, to capture them.
But, and I hate to sound cliche, quantum mechanics might be showing relations beyond this picture of spacetime. Does math and language need to add some new relations to its repertoire to capture entanglement and wavefunction collapse? Or can it get there with its current structure? It kind of depends on what QM really is telling us. But how to get there... If math and language were built from only relations our brain and body could sense, maybe we are in danger of making empirically-refutable mathematics. How would that even look I have no idea. I have come across philosophy of mathematics essays saying math is our most global framework, able to accommodate any and all structure thus far, but still open the possibility of empirical refutation.
I don't want to get too "out there", but QM did spell the end for our classical picture of the world. Math gives us the precise statistics (e.g. wavefunctions), but does not tell us the causal story of QM. There are current-mathematics causal stories of QM (bohmian, etc), but they too disrupt our conception of spacetime and relativity. If the world is not spacetime limited, then doesn't that call into question our senses and language and math if they came about in the above method? I'm not entirely sure how the classical picture fits with math and language, but they seem connected to a large degree.
Re: The Utility of Mathematics
#15This article reminds me of the story of Fourier. It's hard to tell the history of math without the way in which Fourier series and the heat equation shaped the development and increasing rigor in calculus. The mathematicians of the day were extremely skeptical of infinite sums of trigonometric functions because it didn't fit their "a-priori" model of calculus. >Here was the heart of the crisis. Infinite sums of trigo…
This is a curious take on the state of calculus before Cauchy: mathematicians were interested in infinite series, but aware of paradoxes around convergence: for example, the set {1,-1/2,1/3,-1/4,1/5,...} doesn't have a particular sum, but instead you can arrange the elements into series to converge to numbers of any size or even not converge at all. Fourier's work didn't threaten an established notion of calculus, rather mathematicians were having difficulty sorting sense from nonsense in the fertile but chaotic subfield.
It wasn't until half a century later, after Cauchy, that mathematicians had a powerful and coherent foundation for calculus. It's true that then interesting ideas such as inginitesimals were rejected because they lacked comparable rigour: was Bressoud conflating these two time periods?
Re: The Utility of Mathematics
#16Earlier quoted context omitted.
My mind went to Quine as well. One issue is that the empirical world is really some representation of it via human senses, or some consensual agreement on it. I'm not trying to question that there is an external reality, but (along the lines of Quine) one might argue that our understanding of it -- our perception of it, and mental representation of it -- is fundamentally a human representation. One might argue the sa…
Thank you. I've been circling this thought process for a while and didn't know if it is completely incoherent. The thought process began thinking all mathematical equations can be translated to English and vice versa. There must be a shared structure to language and math, or an isomorphism at the least. The effectiveness of mathematics then is more about its conciseness, not that it says anything new about the world…
> The effectiveness of mathematics then is more about its conciseness
Indeed! Ultimately, this is expressed in mathematical notation - which is necessary if you want math to "just work" for you (almost automagically).
> If math and language were built from only relations our brain and body could sense
Which they, of course, are; hence the whole mystery of the "unreasonable effectiveness" of our "everyday" mathematics (such as linear algebra or complex analysis) in areas unreachable to our senses or to our ability even to imagine things (e.g., the quantum world).
Re: The Utility of Mathematics
#17This article reminds me of the story of Fourier. It's hard to tell the history of math without the way in which Fourier series and the heat equation shaped the development and increasing rigor in calculus. The mathematicians of the day were extremely skeptical of infinite sums of trigonometric functions because it didn't fit their "a-priori" model of calculus. >Here was the heart of the crisis. Infinite sums of trigo…
This is a curious take on the state of calculus before Cauchy: mathematicians were interested in infinite series, but aware of paradoxes around convergence: for example, the set {1,-1/2,1/3,-1/4,1/5,...} doesn't have a particular sum, but instead you can arrange the elements into series to converge to numbers of any size or even not converge at all. Fourier's work didn't threaten an established notion of calculus, ra…
Yeah, it's kind of ironic that the modern - and entirely rigorous - direct treatment of infinitesimals comes under the name "nonstandard analysis."
As to the foundations, what Fourier's work shattered was the sufficiency of then established notion of what a function is, which had been limited to what we now call analytic functions; this eventually lead to the abstract definition of a function that we have today.
Re: The Utility of Mathematics
#18> The majority of mathematicians quickly became "Formalists", holding that pure mathematics could not be philosophically considered more than a sort of elaborate game played with marks on paper (this is the theory behind Robert Heinlein's pithy characterization of mathematics as "a zero-content system"). This isn't quite what Formalism is, at least not as Hilbert -- the originator of that philosophy -- described it.…
Quite so. It's also not true that formalism was ever the philosophical viewpoint that most mathematicians felt fit their views best.
Re: The Utility of Mathematics
#19This article reminds me of the story of Fourier. It's hard to tell the history of math without the way in which Fourier series and the heat equation shaped the development and increasing rigor in calculus. The mathematicians of the day were extremely skeptical of infinite sums of trigonometric functions because it didn't fit their "a-priori" model of calculus. >Here was the heart of the crisis. Infinite sums of trigo…
This is a curious take on the state of calculus before Cauchy: mathematicians were interested in infinite series, but aware of paradoxes around convergence: for example, the set {1,-1/2,1/3,-1/4,1/5,...} doesn't have a particular sum, but instead you can arrange the elements into series to converge to numbers of any size or even not converge at all. Fourier's work didn't threaten an established notion of calculus, ra…
There's a lot of historic detail in the book that is left out of this snippet, but his point was that mathematicians barely understood the convergence of infinite series and considered an infinite sum of trig functions to sort of be unfathomable. A big part of Cauchy's work made rigorous fourier series.