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Are there politics in mathematics?
Curious if there are movements within the governance or research pertaining to the field that act to promote or suppress certain ideas? Was watching the “Infinity explained in 5 different levels” and thought… maybe there are trends for or against interpretations and/or abstractions that get a rise in people…
Perhaps the biggest divide is on constructivism. Constructivists reject the law of the excluded middle, which crucially amounts to rejecting proof by contradiction. This is a very useful proof technique, but frequently results in situations where you prove that some object exists but you are wholly unable to describe it or provide an example, which can be morally unappealing.
There are also some historical debates which have been more or less settled (e.g. Is the axiom of choice valid? The prevailing answer is "I don't care," followed closely by "Yes"). In addition, some esoteric views exist which are believed essentially by no one but fervently held by a few, such as ultrafinitism.
Edit: Mathematics also has many different possible foundations (e.g. type theory, category theory, or set theory). The latter-most is the standard, although the former two are heavily researched and are better suited for certain applications (such as automated theorem provers).
"This is a very useful proof technique, but frequently results in situations where you prove that some object exists but you are wholly unable to describe it or provide an example"
You could also discuss mathematics and religion as well as politics.
Since you sound like a learned one in this area, do you think math was discovered or invented? I don't really know enough about math to have an opinion, so if you could give me one to impress other people with I'd appreciate it ha
This is more philosophy than mathematics, and I'm not trained as a philosopher so my answer might not be that articulate.
I think it depends on what in mathematics you're speaking of. Something like an algorithm (e.g. in computer science) or a standard proof technique definitely feels artificial and invented -- it's just something that we as humans use to convince ourselves that something is true or to achieve some goal, and there are many other ways this could have been done, with apparently the same outcome.
But a something like a definition in mathematics feels more discoverable. A good definition ends up in objects which have many satisfying properties and connections to other objects and fields, and this gives off an impression that it was somehow more latent than it seems.
This isn't something that I (or any other mathematician I've met) think of very much though.
Math is just a very specific set of rules and then follow on logic from applying said rules. While the most natural math rules are 1+1=2 to follow the observed world, that doesn't mean there aren't other rulesets that can be used. Abstract algebra is where you apply the common algebraic ruleset to different number systems. It certainly starts to alter your understanding of math. A similar practice can be done by inventing different rulesets and just seeing what happens.
So I would say that math is entirely an invented process. We use the logical process to attempt to describe the natural world, but something to keep in mind is that the natural world does not have to abide. On the surface it does, but the more precisely we try to describe the natural world, the more slippery it becomes.
That's an easy answer even for someone who isn't a mathematician. We are discovering math, we are not inventing it. One and one making two is not something we came up with, we just named it.
Most times I've seen this conversation with real mathematicians involved, the response hovers between two options:
It doesn't matter and few people in math are sweating this.
Both. How we organize our concepts and theories are greatly human driven and invented (and to emphasize this is hard work too and part of math), but the underlying relationships are true regardless of naming, notation or categories and therefore are discovered.
cc:@shieldofv
I brazenly contend: both, although I might prefer the labels natural and artificial. Peeling back some nuances, I don't think the terms are mutually exclusive. Mathematical theory is certainly discovered/natural, but the process of articulating and disseminating it is invented/artificial (not just in the tangible logistic sense, but also in the abstract social sense). If we disregard how humans relate to mathematics from being part of mathematics itself, then whether it's discovered or invented seems inconsequential (to humans) in some sense.
I'm not learned in this area - I just teach 8th grade math as a special ed teacher.
That said, I'm of the opinion that it's both, but mostly discovery. To make water, you need two hydrogen atoms and one oxygen atom. This is one example of how numbers and their relationship to each other are fundamental to our universe. They operate in a definable, predictable way whether we know about them or not.
But we definitely dress math up in a very human way. We are land-dwelling, 3-dimensional beings who experience linear time and primarily experience the world through our sense of sight and sound. This has definitely had a huge influence on our mathematics.
Alien math could be based on touch, smell, or some other sense we haven't imagined. But at the end of the day, we could probably reconcile any differences with enough time and study. Nothing aliens discovered about math would be completely incomprehensible as long as it was all based on the same fundamental experience of reality and a physical world. And even if we did discover aliens who experience time as non-linear and sense the world through echolocation, I think we would find common ground with basic arithmetic and gain a mutual understanding from there without much issue.
This is off topic, but I would have laughed if you used quarks in a proton as your example, as depending on who you ask, it's either three or an infinite number.
I think some models now have 5, too.
Bayesian vs Frequentist statistics come to mind, though I can't pretend to understand what the actual fuss is about. I understand what's going on, I just don't care as much. Maybe because I flip-flop between the two interpretations? Though I think usually I'm more in camp Bayesian.
I'm sure there are legitimate points of contention but the idea just sounds funny to me.
"6 is the best number!"
"No, 4 is superior!"
"Fuck you!"
"You're never getting tenure while I live!"
Bleem is the best number and no one can tell me otherwise.
Way to strawman those of us who recognize the superiority of six. ;-)
Hexagons are the bestagons.
That's a weird way to spell four. Not making fun of you, it is a common typo.
Yes, certainly there is politics like there is in everything, but it tends to be education-related, like about when to teach algebra. Also diversity in the field and what’s fair to students.
A fun example that’s closer to pure math might be the Tau Manifesto?
Mathematics notation seems to result in much less controversy than debates about programming languages (for example). Is that because standardization is less important and commercially relevant for mathematicians than programmers?
Among programmers there is debate about how mathematical we should get. For example, Haskell’s very abstract, mathematical approach is a turnoff for many programmers. This tends not to result in arguments that are all that heated, though.
I'm sure since humans are involved, there are plenty of local examples where emotions, opinions and competition got in the way. I could also see another aspect of being human get in the way, pride. If you spend your whole life with a set of views and those get crushed, you may still refuse to accept the evidence.
So if the head of any institute with some authority about who gets funding or who gets published has certain views, those may get promoted.
In physics it seems to be more visible though - I don't think the popularity of string theory is not just based on scientific merit. David Gross said at some point that one of the reasons why the theory is so popular is 'there are no other good ideas around.' That is definitely a biased statement that seems more political than scientific to me.
To add on to this, even though it goes beyond OP's somewhat narrow definition of what "politics" means: there is certainly a lot of fighting over the scarce rewards in academic mathematics. An example that comes to mind is that of (https://en.wikipedia.org/wiki/Grigori_Perelman#Possible_withdrawal_from_mathematics)[Grigori Perlman], the person who solved one of the Millenium prize problems by proving the Poincare conjecture. He famously rejected the prize money, as well as equivalent of the Nobel prize for mathematics. He eventually quit research mathematics altogether in part due to his perception of poor ethics even among very prominent mathematicians, who were more than willing to downplay his contributions to the proof of the Poincare conjecture in order to raise the profiles of their students.
So I'd wager that the "politics" will likely never be about what is mathematically true. Any "suppression" that OP alluded to will likely come in the form of which research programs/grants get funded and which ones don't. The jockeying for a piece of a very small pool of money will depend equally much on the quality and the trendiness of the research, as well as on nonscientific aspects like connections and influence. Despite how pure of a pursuit mathematics seems, it, and all other fields, will never be free from human desires for money, fame, and lasting legacy.
Yeah, this is a sad aspect of how academic funding works. A friend of mine is a pretty successful researcher who has secured his own funding through hard work and grit -- but he still laughs about how his personal income is peanuts compared to folks who've went into industry.
Physics has a lot more money involved than mathematics so it's unsurprising politics enters into it more.
While not my expertise, I am a "hobbyist" here, and if you're interested in quantum gravity, even if you think strings are wrong, it's the most developed and so far fruitful game in town. So he's not entirely wrong here.
With that said, pop science cheerleaders for String Theory have done, imo, more harm than good. It's a little frustrating to see normal folks group ST, Dark Matter, and Dark Energy together as science gone wack.