Towing the Lion

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Warren
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Re: Towing the Lion

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Acyrologia? That doesn't sound anything like mallet-prop.
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Re: Towing the Lion

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The American Traditional tattoo design style usually lacks a lot of complicity since it was done years ago with less advanced equipment and the tattoos were made fast.
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Re: Towing the Lion

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Ellie wrote: 10 Mar 2021, 23:27
The American Traditional tattoo design style usually lacks a lot of complicity since it was done years ago with less advanced equipment and the tattoos were made fast.
American Traditional is nothing if not law abiding.
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Re: Towing the Lion

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Image
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Re: Towing the Lion

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Free yourself from addition forever, major in the liberal arts.
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Re: Towing the Lion

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Hugh Akston wrote: 06 Apr 2021, 00:47 Free yourself from addition forever, major in the liberal arts.
I think you mean humanities.
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Re: Towing the Lion

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lunchstealer wrote: 06 Apr 2021, 01:29
Hugh Akston wrote: 06 Apr 2021, 00:47 Free yourself from addition forever, major in the liberal arts.
I think you mean humanities.
Both include mathematics.
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Re: Towing the Lion

Post by Warren »

D.A. Ridgely wrote: 06 Apr 2021, 02:59
lunchstealer wrote: 06 Apr 2021, 01:29
Hugh Akston wrote: 06 Apr 2021, 00:47 Free yourself from addition forever, major in the liberal arts.
I think you mean humanities.
Both include mathematics.
lol
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Re: Towing the Lion

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D.A. Ridgely wrote: 06 Apr 2021, 00:33 Image
I guess that sums it up.
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Re: Towing the Lion

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Warren wrote: 06 Apr 2021, 11:50
D.A. Ridgely wrote: 06 Apr 2021, 02:59
lunchstealer wrote: 06 Apr 2021, 01:29
Hugh Akston wrote: 06 Apr 2021, 00:47 Free yourself from addition forever, major in the liberal arts.
I think you mean humanities.
Both include mathematics.
lol
Warren, you are if, nothing else, predictable. But of course mathematics, per se, is among the humanities. It's not science, physical or social. Applied mathematics is an important, I'd even gladly allow that it's the most important field of mathematics, but it's not mathematics, per se. And whatever you did for a living using mathematics with your electrical engineering degree and however much you would like to use it to prove your intellectual superiority to English majors in college and such because you can solve calculus problems and they can't, you weren't doing mathematics, you were just using it. You were calculating. And, sure, figuring out what calculations were necessary to arrive at an answer to an empirical question, which puts what you were doing a step above being a mere mechanical calculator but not much closer to being a mathematician.

What Jadagul does for a living, teaching differential equations to engineers aside, has more to do with music and art than it does with biology or electrical engineering. Mind you, that's not to say that scientists and even engineers don't occasionally do mathematics in the sense I mean here. It's not to say that they're less intelligent than mathematicians or, for that matter, that you couldn't have been a mathematician if you had wanted to be. It's also sure as hell not saying that engineering doesn't require creativity and intelligence, let alone that the physical sciences don't. Of course they do. But in general they're using math, not doing math. Theoretical physics blurs the line, but on balance physical sciences use but don't do mathematics.

Now, all that said, I'm going to take it all back. If you want to say mathematics belongs with science and engineering insofar as universities divvy up all of their disciplines, that's fine with me. There are no facts that make or break my case and there are no facts that make or break your case. It's ultimately a silly dispute not all that far removed from whether you prefer chocolate or vanilla. I've given an argument for why I think the way I divvy up academia makes more sense. You can continue to scoff, you can ignore the argument or you can offer a counter-argument, but in the end all of those arguments back and forth cannot result in a definitive resolution of the question. It isn't empirical and there are no formal proofs available to determine the truth here. So it's a matter of persuasion. Do I believe I'll persuade you? Of course not. But I might just get other people to see the question from a new perspective and, frankly, that's the only reason I bothered with this.
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Re: Towing the Lion

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I have long maintained that math is part of the humanities, unless you're a funding source, in which case we're the science-iest of the sciences.

Realistically I think grouping math institutionally with the sciences makes sense, because all the people doing the sciences need a ton of math and the people doing history mostly don't (but see my hypothetical book arguing that thinking like a mathematician would help everyone!).

But math isn't really a science and is philosophically more in tune with the humanities (and often specifically philosophy). It's also a rhetoric-focused discipline, though the rhetorical rules are so weird that most people often don't realize that it is that. There's a reason "STEM" has a separate letter just for "math".
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Re: Towing the Lion

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D.A. Ridgely wrote: 07 Apr 2021, 00:01 Applied mathematics is an important, I'd even gladly allow that it's the most important field of mathematics, but it's not mathematics, per se.
You go too far.

Understanding the properties of differential equations, especially beyond the level of the sophomore courses taken by philistine engineers, is deep and beautiful and tied to both practical and pure issues. Information theory is both eminently practical and a source of profound insights. Ditto the theory of computation. Number theory both undergirds the cryptography that enables modern digital commerce and addresses fundamental questions that go back to Greece.

I will freely grant that the courses taught to the masses of engineering and science majors emphasize calculation over insight, but for most all of those tools one need only take a short step to see profound things. And the applied mathematicians who study these things beyond the bare level needed to teach freshmen and sophomores are real mathematicians.
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Re: Towing the Lion

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BTW, given the importance of applied math in commerce and public administration through the ages, math should be of as much interest to historians as irrigation, transportation, metallurgy, and other technological developments. I realize that emphasizing the significance of math for history and society does not a priori mean applied math is a humanistic discipline in the way that pure math is, but it does mean that applied math is intertwined with human endeavors to the point where we cannot understand our own progress without it. A truly broad, humanistic education needs applied math to understand society as much as it needs pure math to understand issues issues in philosophy and logic.
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Re: Towing the Lion

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thoreau wrote: 07 Apr 2021, 00:39
D.A. Ridgely wrote: 07 Apr 2021, 00:01 Applied mathematics is an important, I'd even gladly allow that it's the most important field of mathematics, but it's not mathematics, per se.
You go too far.

Understanding the properties of differential equations, especially beyond the level of the sophomore courses taken by philistine engineers, is deep and beautiful and tied to both practical and pure issues. Information theory is both eminently practical and a source of profound insights. Ditto the theory of computation. Number theory both undergirds the cryptography that enables modern digital commerce and addresses fundamental questions that go back to Greece.

I will freely grant that the courses taught to the masses of engineering and science majors emphasize calculation over insight, but for most all of those tools one need only take a short step to see profound things. And the applied mathematicians who study these things beyond the bare level needed to teach freshmen and sophomores are real mathematicians.
All true. That said, all applied mathematicians are mathematicians but not all people who use applied mathematics are mathematicians.

And even that's unfair to many people. You've done philosophy; I've read you doing it here. So, for that matter, have most of us. Doesn't make us academic philosophers, to be sure. But anyone who grasps Turing machines relationship to computers is in that moment thinking like a mathematician. Anyone who sees that Einstein's work depends on non-Euclidean geometry but non-Euclidean geometry has only a fortunate but accidental relationship to physics is thinking like a mathematician. Anyone who grasps that, insofar as we understand infinity to be a number and/or a set, there are some infinities larger than other infinities, our intuitions be damned, is thinking like a philosopher and a mathematician. And so on. But in the academy what mathematicians do is, applied mathematicians aside, a humanistic endeavor indifferent to any 'real world' applications. That's the sword I'll fall on.

But, yes, also I agree with your follow-on post.
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Re: Towing the Lion

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thoreau wrote: 07 Apr 2021, 00:39
D.A. Ridgely wrote: 07 Apr 2021, 00:01 Applied mathematics is an important, I'd even gladly allow that it's the most important field of mathematics, but it's not mathematics, per se.
You go too far.

Understanding the properties of differential equations, especially beyond the level of the sophomore courses taken by philistine engineers, is deep and beautiful and tied to both practical and pure issues. Information theory is both eminently practical and a source of profound insights. Ditto the theory of computation. Number theory both undergirds the cryptography that enables modern digital commerce and addresses fundamental questions that go back to Greece.

I will freely grant that the courses taught to the masses of engineering and science majors emphasize calculation over insight, but for most all of those tools one need only take a short step to see profound things. And the applied mathematicians who study these things beyond the bare level needed to teach freshmen and sophomores are real mathematicians.
"Practical" isn't the same as "applied".

As "applied math" is understood within the discipline of mathematics, arguably none of the things you just listed are applied math.

It's "applied math" if you're using it to solve problems in other disciplines. So if your research is in modeling biological systems, or implementing cryptographic protocols, you're in "applied math".

If you are studying the properties of differential equations, or information theory, or elliptic curves, then you're not really in applied math. I remember saying that my undergrad PDEs professor was an applied mathematician and he kind of double-took and told me that _he_ didn't think he was an applied mathematician, since he did PDE theory.
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Re: Towing the Lion

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Jadagul wrote: 07 Apr 2021, 01:48 "Practical" isn't the same as "applied".

As "applied math" is understood within the discipline of mathematics, arguably none of the things you just listed are applied math.

It's "applied math" if you're using it to solve problems in other disciplines. So if your research is in modeling biological systems, or implementing cryptographic protocols, you're in "applied math".

If you are studying the properties of differential equations, or information theory, or elliptic curves, then you're not really in applied math. I remember saying that my undergrad PDEs professor was an applied mathematician and he kind of double-took and told me that _he_ didn't think he was an applied mathematician, since he did PDE theory.
That's fair, though the lines are surely fuzzy. "Does this system of equations have solutions with properties similar to these real world phenomena?" is a question about the equations. "Does this algorithm do a good job (by some precise criterion) of pulling this type of information from noisy data?" is as much a question about the algorithm as the application.

I think a lot of it comes down to how you write the last few paragraphs of the paper. If your conclusion is that adding a certain type of term to the equations is necessary to get solutions with behavior matching these observations, you're doing math. If your conclusion is that these equations imply something about the real world and someone should do observations to check that, you are doing applied math.
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Re: Towing the Lion

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I read this a few weeks ago, and I couldn't decide if "intuitive math" was pitching woo, lazily calling an estimate good enough or, as being described here, doing math-as-philosophy.

https://www.quantamagazine.org/does-tim ... -20200407/

I never did more than a few philosophy classes but the "proofs" reminded me of math but always struck me as fuzzy enough to make me think "really? That's considered "proof"?
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Re: Towing the Lion

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dead_elvis wrote: 07 Apr 2021, 15:11 I read this a few weeks ago, and I couldn't decide if "intuitive math" was pitching woo, lazily calling an estimate good enough or, as being described here, doing math-as-philosophy.

https://www.quantamagazine.org/does-tim ... -20200407/

I never did more than a few philosophy classes but the "proofs" reminded me of math but always struck me as fuzzy enough to make me think "really? That's considered "proof"?
Intuitionist math is actually none of those; the name sucks. There's a sense in which intuitionistic math is 'more' rigorous, because it restricts the things you can do more harshly than 'normal' math does. (I myself would say they're equally rigorous, though, because they both follow their own rules carefully.)

The basic idea is that you don't want to let people say "there exists a thing" without explaining where you find it. So it rules out things like the intermediate value theorem, that says "if f is continuous, and f(a) <0 and f(b)>0, then there must be some number c with f(c)=0".

The explanation in the quanta article is a bit confused, well below their usual standard of clarity in math articles. But intuitionism is kind of confusing and I struggle with it myself sometimes, so that maybe shouldn't be surprising.
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Re: Towing the Lion

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thoreau wrote: 07 Apr 2021, 12:13
Jadagul wrote: 07 Apr 2021, 01:48 "Practical" isn't the same as "applied".

As "applied math" is understood within the discipline of mathematics, arguably none of the things you just listed are applied math.

It's "applied math" if you're using it to solve problems in other disciplines. So if your research is in modeling biological systems, or implementing cryptographic protocols, you're in "applied math".

If you are studying the properties of differential equations, or information theory, or elliptic curves, then you're not really in applied math. I remember saying that my undergrad PDEs professor was an applied mathematician and he kind of double-took and told me that _he_ didn't think he was an applied mathematician, since he did PDE theory.
That's fair, though the lines are surely fuzzy. "Does this system of equations have solutions with properties similar to these real world phenomena?" is a question about the equations. "Does this algorithm do a good job (by some precise criterion) of pulling this type of information from noisy data?" is as much a question about the algorithm as the application.

I think a lot of it comes down to how you write the last few paragraphs of the paper. If your conclusion is that adding a certain type of term to the equations is necessary to get solutions with behavior matching these observations, you're doing math. If your conclusion is that these equations imply something about the real world and someone should do observations to check that, you are doing applied math.
Honestly, that's pretty much it. It gets billed as "applied math" if you're actually working with a real world situation and real world data and giving advice to doctors and engineers. If you're just talking about the behavior of the equations it's generally more-or-less pure math, at least we tend to classify it.

So if you write a paper about fractional differential equations, you're a pure mathematician. If you write a paper about how you can use fractional differential equations to study groundwater contamination, you're an applied mathematician.
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Re: Towing the Lion

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When I worked on tumor models, I actually thought that the physicists were better mathematicians than the applied mathematicians. What I mean is that the applied mathematicians would show up to the conference with some model that has 20 parameters and then show that the solutions to those equations qualitatively resembled the real-world phenomena. The physicists would argue that the solutions to the equations should be relatively insensitive to whatever parameters, so all we need to focus on is this tiny handful of parameters, which control all the interesting behavior.

Yes, the physicists arguments would have fallen short of what a pure mathematician would want, but the spirit was much more looking at unifying structure behind the possible models, while the applied mathematicians were excited to throw 20 parameters into their models.

Needless to say, the biologists much preferred the applied mathematicians, because biologists have zero appreciation for abstraction.
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Re: Towing the Lion

Post by D.A. Ridgely »

Jadagul wrote: 07 Apr 2021, 18:02
dead_elvis wrote: 07 Apr 2021, 15:11 I read this a few weeks ago, and I couldn't decide if "intuitive math" was pitching woo, lazily calling an estimate good enough or, as being described here, doing math-as-philosophy.

https://www.quantamagazine.org/does-tim ... -20200407/

I never did more than a few philosophy classes but the "proofs" reminded me of math but always struck me as fuzzy enough to make me think "really? That's considered "proof"?
Intuitionist math is actually none of those; the name sucks. There's a sense in which intuitionistic math is 'more' rigorous, because it restricts the things you can do more harshly than 'normal' math does. (I myself would say they're equally rigorous, though, because they both follow their own rules carefully.)

The basic idea is that you don't want to let people say "there exists a thing" without explaining where you find it. So it rules out things like the intermediate value theorem, that says "if f is continuous, and f(a) <0 and f(b)>0, then there must be some number c with f(c)=0".

The explanation in the quanta article is a bit confused, well below their usual standard of clarity in math articles. But intuitionism is kind of confusing and I struggle with it myself sometimes, so that maybe shouldn't be surprising.
For anyone who cares to go down that particular rabbit hole, I'd recommend the Stanford Encyclopedia of Philosophy article on Philosophy of Mathematics, at least the first part of which is fairly intelligible. While Intuitionism is a particular school of philosophy of mathematics apparently enjoying renewed popularity as in that linked article, I don't know whether it counts as mathematics, per se. I'd of course defer to Jadagul on that because basically mathematics is whatever mathematicians do and whatever they say it is. I'm not responding to his last comment, I'm just commenting on the topic a bit further.

My hot take is that there is a sense in which what a mathematician accepts as mathematics or foundations of mathematics is in some way ultimately a matter of intuition in its ordinary sense or, to lift a quote from Rorty, whatever you can get away with. For logicians (some of whom are mathematicians, some of whom are philosophers, some of whom are both) there are arguments in and about formal logic, for example, as to what to accept as axiomatic, and it seems to me that such disputes do come down to a matter of intuition.

Most academic philosophers accept as both rigorous and intelligible first order logic a.k.a. first order predicate calculus; it's what a grad student would have to have some reasonable grasp of for preliminary exams. Beyond that, what sense to make of modal logic or whether one accepts the law of the excluded middle, whether infinity is a set or a number or both (Wittgenstein said, not entirely joking, that "infinity" just meant you can keep going), etc. are way above my philosophical competence; I don't even understand most of this stuff well enough to have much of an opinion.

Just to add to the confusion, "intuition" as a concept applied to mathematics goes back at least to Plato's Meno dialogue in which (using Socrates as a sock puppet) he purports to get an unschooled boy to grasp some geometric inferences via what Plato calls anamnesis, i.e., a 'recollection' of understanding of the Forms. Kant's use of the term intuition is roughly that our minds process sensations through two fundamental 'pure' intuitions: space and time, which are respectively responsible for our grasp of phenomena in space and of time.
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Re: Towing the Lion

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That all seems right.

In this case, "intuitionism" is a different take on what rules of logic we're allowed to use while doing mathematics. But the thing is, while "we ought to use rules XYZ" is a normative philosophical position, "if we use rules XYZ then we can prove A but not B" is a positive mathematical assertion.

So like, most mathematicians accept the "normal" rules, and are comfortable using infinite sets and non-constructible numbers and the law of the excluded middle etc. And a lot of the intuitionists or constructivists or finitivists will claim that that's "wrong". But the people who are fine with the Law of the Excluded Middle can still accept "I proved Theorem A without using LEM" as a valid mathematical statement and proof.

And thus while arguing about what rules we should use is largely pre-mathematical, working out the implications of any given set of rules is fully within the bounds of mathematical research.
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Re: Towing the Lion

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Really, it just bugs me because I deal with a lot of English learners who struggle greatly with forming questions in English (which is not an easy matter) and this kind of thing does not help.
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