Lately, there have been many people who say that mathematics itself is racist, that it is simply a creation of dead white Greek men. As a mathematician, I strongly disagree, and believe that mathematics transcends any particular culture or political system. For example, while the Arabs invented algebra, algebra is not itself Arabic, but universal. Perhaps, however, as a mathematician I am simply biased. That raises the question, is mathematics politically and culturally neutral? I would love to hear good arguments that challenge my position that mathematics is apolitical and acultural.
16To answer the question, it would be helpful to see the actual, specific claim being made. To me it's difficult to imagine that mathematics in itself is racist. However, there are definitely ways to use it to push an agenda. Infamously, in Nazi Germany, there were textbook problems along the lines of "A cripple costs the State 5,5 Reichsmark per day. How many young familiy loans at 1000 Reichsmark could be given out if we got rid of all the 200.000 cripples in the Fatherland?" While not technically racist, I think the message is clear.– MookuhSep 19, 2022 at 7:20
1Related questions were explored by Shulman (1996, Configurations **4**(3):427-451), but I don't think she reached any conclusion suitable for the SE Q&A format.– Daniel HattonSep 20, 2022 at 12:00
Maybe the issue is your choice of words : Arabs didn't INVENT algebra, but just discovered/understood it ? It's not something they woke up some morning and decided "Hey, now things work that way". It always worked that way, they just managed to get it first.– MoukeSep 20, 2022 at 13:39
1In pre-WW2 USSR quantum theory, relativity, and other contemporary scientific developments were considered borgeois/capitalist science, contradicting Marxism-Leninism, and people went to Gulag for trying to learn/teach these subjects. Later, during the war and the Cold War, this point of view was revised for obvious practical reasons, and physics and exact sciences became fields where people were allowed some free through (hence the significant number of dissidents). However, this attitude continued in Genetics, and some other fields, like geology... and obviously in economics. (contd.)– Roger VadimSep 20, 2022 at 14:24
1(Contd.) See, e.g., Lysenkoism. There are also historical reasons why one would want to oppose certain subjects - You may well think that the facts that math was developed by Greeks and Arabs have nothing to do with its content... but for many people these are the only things that they know about math.– Roger VadimSep 20, 2022 at 14:25
From its inception, mathematics was intended to be independent of cultural contexts in the real world. For example, you could use the same rules of algebra and the same numerals to count your family members, your neighbors, olive trees, goats, boats, etc. There is nothing racist about possessing the ability to equitably divide one's goats between one's children, is there?
And you are exactly correct to point out that there is no "Greek mathematics", no "Roman mathematics", no "Indian mathematics"- they may all use different symbols to represent quantities and operations, but if you present a herd of goats to a Greek, a Roman, and an Indian mathematician and ask them to count the goats, they will all furnish the same answer.
4No holy wars are fought over arithmetic. Sep 18, 2022 at 0:34
19@ScottRowe: But Pythagoreans thought discovering irrational numbers was worthy of being murdered by the gods, use of Arabic numerals for accountancy was resisted in some European countries until as late as the 1800s as suspect, Newton insisted on the superiority of geometry and geometrical proofs as the realist godliest math, and some people still get very het up over the ontological status of infinity. I'd say there's still time. Sep 18, 2022 at 0:58
18OP was specifically asking for arguments against his own position, and this answer does nothing but reinforce it. Why OP accepted it puzzles me. Note that I'm not suggesting there are good arguments supporting math is culturally biased, I don't know of any. Just reminding this is what OP was looking for.– armandSep 18, 2022 at 6:45
5@CriglCragl Do you have reliable sources for these 3 claims? Sep 18, 2022 at 17:51
5This answer started well with "mathematics was intended to be independent of cultural contexts" - even emphasising "intended", but then it unfortunately missed the fact that one doesn't always achieve what one intends to achieve, and humans are unable to step outside their cultural context, therefore they're unable to fully escape it, regardless of which field we're talking about. Sep 19, 2022 at 0:36
Yes and no. Who writes papers and what they choose to research are products of culture, and in the US, our culture has a long history of racism. The actual theorems in the papers themselves are generally not reliant on any racial theory.
Math models reality in precise, abstract, and formal terms. The idea of thinking about the world in precise, abstract, and formal terms is not exclusive to one part of the world or one ethnic group, it has been developed independently in many places and there has been a lot of cross-cultural collaboration across the centuries.
As an idea, it predates "whiteness" by several thousand years. To take an example, Pythagoras was probably a cultural chauvinist, but it is extremely unlikely that he was racist in any sense we would recognize. He would have had no cultural conception of "whiteness" as we understand it (it hadn't been invented yet), and we have absolutely no way of knowing how dark his skin was. There is no sense in which we can say that Pythagoras is a dead white Greek man.
However, geometry is still a cultural construct and we can't escape that. Pi is undisputedly the ratio of a circle's circumference to its diameter, but a geometric circle is itself a complex concept that a culture develops over time. The Pythagoreans who developed this idea had certain cultural prejudices which led them to develop geometry rather than algebra.
I learned about irrational numbers in the context of the Pythagorean school. The concept was also developed in India, in a different, more algebraic context, but I didn't learn that as a child. Math instruction--what concepts we introduce, what order we introduce them, and what context they are given--is cultural.
So in the sense that much of modern math is the product of a research agenda set by racist power structures, and is taught in a cultural context with--to put it mildly--a lot of racial baggage, math does reflect racism, and to some degree is even the product of racism. That does not make math a racist endeavor by nature, and it does not mean that math itself is racist.
I can draw an analogy with highways. It is a fact that in the United States, racism heavily influenced where highways were built and who benefited the most from them. That doesn't make roads racist. A road is just a big trail. Roads aren't racist. Building roads is not racism. Forcing residents of a black neighborhood to sell their houses to you so that you can build a road to benefit a white neighborhood is racist. The resulting shape of a city, after all those roads are built, reflects racism.
Princeton University was, in part, constructed by slaves and funded by slaveowners. During much of its storied history, Princeton did not admit black students or women. To this day, both historical and modern racial politics affect the day-to-day life of a student at this top-ranked school of mathematics. Does this make any math produced by a Princeton researcher racist? No. However, in the absence of blind peer review, we can reasonably ask whether that paper would have been published if the author had come from Delaware State.
Writing a mathematical paper is not racist. An abstract proof is not racist. A college building that you sat in to write the paper is not racist. Having a building built is not racist. Forcing "black" people to build a college for the benefit of "white" people and then building generations of white wealth on that labor is racist. The resulting shape of mathematical research reflects that racism.
So my answer would be both "yes" and "no". Mathematical theorems are not racist. The act of thinking about the world in precise abstract terms is not racist. But the larger mathematical endeavor does take place in a cultural context of racism.
20"a circle is itself a cultural construction": say what?– KazSep 19, 2022 at 5:56
9I am confused about what you say about circles. It defines a specific geometric form. How is a geomectric form a cultural stuff ? Sep 19, 2022 at 7:21
13"our culture has a long history of racism (although ancient Greece did not)" Lol. Aristotle believed in ethnicities that were 'natural slaves', Plato that citizenship should only be by inheritance. Ancient Greece was wildly racist, denying citizenship to immigrants and even most of it's people that left to found colonies, & believing their people had special qualities because they had chthonically sprung directly from the local soil. Sep 19, 2022 at 9:36
15A circle is a shape that's found in the natural world. It's as much of a cultural construct as an elephant is (which is to say: we subjectively interpret the objective reality and label things as circles and elephants, and our subjective experience and labels may depend on culture). But we (at least most of the internet-using population) can all agree what a circle and what an elephant is (with probably some disagreement over what a species is), so perhaps not the best example of how maths isn't "politically and culturally neutral" and may be racist. Sep 19, 2022 at 9:52
11Roughly circular shapes occur in nature, but our notion of a circle as a type of object in Euclidean geometry is a cultural construct: we developed a culture in which Euclidean geometry was developed, propagated, and applied, and to people who are not accustomed to Western ways of abstract thinking (such as all small children) this conceptual framework can be difficult to grasp. Sep 19, 2022 at 11:04
A somewhat different perspective can be gained by taking a close look at the media of mathematical expression across history.
For example, the Bourbaki school of mathematics shifted the language and culture of mathematics away from visual and linguistic argumentation in favour of symbolic manipulation. Arguably this led mathematical research to privilage certain directions of research such as analysis, abstract algebra, and point-set topology over other directions such as traditional (synthetic) geometry.
Later in the 20th century the rise of category theory brought diagrammatic reasoning back into the culture of higher mathematics, but to this day in many mathematics departments category theory is not regarded as sufficiently legitimate or prestigious to hire a candidate based on being a category theorist alone. Moreover, a student of mathematics is unlikely to encounter category theory or anything that makes extensive use of diagrammatic proof techniques until quite late in the educational process (i.e., not until grad school). I think it's likely that many students who might have excelled in visual proof would be weeded out of math before they ever find out it is an option.
To go further back in history. Before the notational innovations of the the Renaissance, mathematical proofs were written in full prose text. This mode of expressing mathematical thinking would have made people with higher linguistic intelligence more likely to excel as mathematicians. In contrast, today's heavily symbolic notation likely makes people with high calculative intelligence more likely to excel as mathematicians. Today people with higher verbal or linguistic intelligences are generally shunted towards the humanities. Similarly, people with high visuospatial intelligence will do better in mathematical cultures that use drawings and diagrams than they would in the post-Bourbaki culture of the mid-20th Century.
Turning now to today's practices of mathematical research. It's striking to me that in spite of the fact that with computer technology we could make use of colour-coding in mathematical notation, this is almost never used in research mathematics. This contrasts with established practices in software development where colourful syntax highlighting is the norm. This choice to keep mathematical notation black-and-white is not gender-neutral since women on average have higher abilities for colour perception and classification.
The dominant modes of expression for mathematical thought today strongly emphasize symbolic shape (glyphs), sequence, and relative spatial positioning. Transitions in meaning are expressed through re-inscription of symbols with permutation. These features of mathematical expression favour a certain profile of perceptual and cognitive abilities. Before computers with rich graphical displays it might have been argued that this was the only practical way to express complex mathematical ideas, but that is no longer the case today. Insofar as the profile of cognitive-perceptual abilities might favour certain groups over others this represents a bias in the mathematical practice of our culture which privileges certain groups over others. People who are "good at math" in this culture are, among other things, those who are particularly good at visually tracking and mentally manipulating black-and-white symbols. If the capacity for abstract thought is orthogonal to the capacity to excel in the use of the means of expression of modern mathematics notation, then this could represent an injustice. It at least calls for more investigation.
3+1 for the idea of color use in math notation. I have been thinking about it, it would be nice to see more developments ... Sep 18, 2022 at 21:51
9as someone who is colorblind, i find it anathema that the ability to read and study mathematics could perhaps be made inaccessible to any with this physical disability. mathematics is only black and white on the printed page; otherwise, it’s about wrestling with and playing highly abstracted concepts that somehow rather oddly tells us things about the world we live in with all its vicissitudes and color too!– A.EllettSep 19, 2022 at 0:08
6@A.Ellett That's a great point. The nice thing about syntax highlighting is that it can easily be personalized to one's own preferences. In any case, the example of using colours in math notatation was not the main point of my answer. What you point out about the possibility of colours making math notation inaccessible reinforces the broader philosophical argument I was making. That the modes of expressing mathematics are not neutral or universal and they have consequences for who is likely to excel in mathematics.– Avi CSep 19, 2022 at 1:15
3As someone who has seen a good deal of category theory and diagrammatic proof techniques I have no idea why you associate them with visual proofs. I could see an argument that most math notation uses a single direction, namely left to right. Some cultures write top to bottom or other directions. Diagram chasing makes full use of 2 dimensions at once. This is an interesting distinction but I wouldn't call it visual.– quaragueSep 20, 2022 at 9:24
1@kjetilbhalvorsen This could be an interesting conversation to have, but probably not in comments here. Nevertheless, color-blindness isn't a straight-forward & clear-cut matter. For myself, colors that I can distinguish in broad strokes might be indistinguishable to me if in a finer point. And it's not as simple as saying one's color blindness is red-green or something else. Yet, I can see how colors are informational in cartography and the presentation of certain graphs, but then those generally are not things I can reliably use or understand.– A.EllettSep 20, 2022 at 16:25
It would have been much easier to respond with a more focused answer with specific examples of the claims you're wanting to engage with. Most of the fuss I can see around the topic seems to center on this controversial textbook or the occasional comment about history so I'll prod at some of that.
This seems like an important distinction to start with: the teaching of mathematics entails additional cultural context. For an overly obvious illustration, "8-5=3" isn't racist but if you have a word problem like "Sally has $8 and Jamal steals $5. How much is left?" it becomes clear that there's more encoded in the question than just the maths! This suggestion comes back when people complain about named theorems; just as the former question implicitly codes Jamal as the thief, talk of Hamiltonians and Jacobians (without similarly prestigious names of Africans for example) codes the white guy as the mathematician. This of course is not at core a problem with maths: matrices of partial derivatives would behave the same whatever name we gave them. It's really a problem of history and language. But it still behooves us to reflect on the cultural conventions of how we teach or communicate those statements.
One specific issue which the textbook discusses and the responses raise as a concern is the handling of objectivity. Objectivity itself is listed as one of the markers of the "white supremacy characteristics." Why is out of scope, but suffice to say this causes no end of consternation among respondents who insist that mathematics is just objectively objective. 8-5=3, and that's that. It should be noted that even this text is not saying, as some might insinuate, that to avoid being racist we should allow for the opinion that 8-5=29. Instead, it is suggesting that where there is space for value judgement in interpreting the problem and applying the solution, we tend to push people to one right answer instead of training them to make those judgements. For example, consider a traffic routing problem. We'd teach people to throw some graph theory at it, and give the marks if they provide the shortest, fastest route. Actual engineers writing mapping algorithms, however, have to consider value judgements like whether to prioritise the individual or the collective. They need to consider questions like "What if everyone takes this route (e.g. because everyone uses this algorithm) and so we congest the road?" That is, there is a decision about what to optimise for which goes beyond the objective core of mathematics.
This principle that the objectivity stops where the application starts carries beyond the classroom. Because mathematics has this aura of objectivity, it is used to lend objectivity to other statements. You might hear something like "I'd love to give you a raise, but I can't. It's just maths." That has an air of finality to it, whereas a mathematician trained in evaluating applications might peel back the covers to see "... but I can't if I want to keep two sigma below my expected profit margin above what my investors would get in a savings account." It's easy to see how the former understanding of mathematics gives cover to societal oppression, whereas the later opens up negotiation of trade offs.
All this is to say that the core of mathematics is not racist. The rules of arithmetic or calculus or probability or group theory don't favour any particular group of people. Although axioms they cannot, strictly speaking, make the same claim to objectivity, even mathematical controversies like whether to take the axiom of choice are not where things get difficult. Notwithstanding all this, discussion and application of mathematical facts happens in a certain social and cultural context, and it is wise to be aware of that.
2Good answer, but I think there are far more direct implications if we poorly apply the principle of "pushing people to the one right answer": students learn that they need to do everything exactly as instructed and come up with the right answer in the method prescribed, there's no room for thinking for themselves, and if they can't come up with the right answer, they shouldn't even bother. Of course this will greatly hurt their ability to solve more subjective real-world maths problems, but this is deeply counter-productive even for the "objectively objective" maths problems. Sep 19, 2022 at 0:16
5I'd like to comment on the part about the names of the theorems. As an Italian, I'm used to calling theorems by what they prove, but I've found during and after university that the English-speaking world (or at least the US) behave differently. For example, it took me a while to understand what the "Nyqvist theorem" was, because I was used to calling it "the sampling theorem". Granted, we still call some theorems by the names of their authors (in high school we say "Lagrange theorem" more commonly than "average value theorem", though we're taught both). More of a cultural difference, I guess.– SimoneSep 19, 2022 at 9:03
Well, many students see early on the Rao-Blackwell theorem. Read about David Blackwell here: en.wikipedia.org/wiki/David_Blackwell Sep 20, 2022 at 12:29
1@Simone Scientific units are often named for people. In electronics we have Amp, Farad, Coulomb, Volt, Henry and so on. But it does make it easier to associate the units with the history. Maybe all names should be dropped from history? Most of those people are long dead anyway. Sep 20, 2022 at 15:49
1@kjetilbhalvorsen Not to mention Noether's theorem, the Ramanujan conjecture, Yang-Mills theory, the algorithm, etc. Sep 20, 2022 at 17:35
Many of the answers before this give good reasons to believe that mathematics is neutral.
Here's an argument for why it might not be neutral. While mathematics itself is neutral the decision of WHAT mathematicians choose to study, what gets published, what receives awards/funding, what is considered deep/profound and not profound are certainly biased culturally.
Countries with more money to spend on mathematics research get a stronger say in what research is considered worth doing. People living in those countries get biased as they go through their education about what is considered "deep" and "profound" and find that their interests are shared by the wider world. The other way around might not be true.
I suppose computer programming as a career is heavily influenced by "where the money is" in the available jobs. Sep 24, 2022 at 21:12
This answer contains many assertions but zero examples or evidence. Oct 1, 2022 at 3:02
Whether a form of mathematics is logically sound doesn’t have anything to do with culture, and some mathematical discoveries have been either made independently or accepted by every culture. Some things that are culturally-dependent include:
- What mathematical topics and historical mathematicians are considered most important
- What arbitrary notations we use (Left-to-right with Greek and Latin letters, versus right-to-left with symbols in Arabic script, for instance)
- What style and techniques we prefer for proofs and textbooks
- Which system of axioms to work from, potentially, although I’m not aware of there being a divide between some countries preferring ZFC, constructive mathematics, or so on, to an extent that ignoring their preferred system of mathematics would constitute bias.
- What vocabulary to use to discuss mathematics, particularly when there are differences in dialect
- What examples and applications to use to keep students interested
You don’t give examples of what criticisms you’re referring to, but some people describe any flawed method of teaching that puts racial minorities at a disadvantage as a form of systemic racism, and it is certainly conceivable that some other approach would be better at closing the racial gap in math scores. More controversially, you see some writers arguing that the gap is partly an illusion, and we simply need to become better at measuring what underrepresented students are already learning and recognizing it as equally valid. That is much more difficult to assess, and might not be relevant if our goal is to get more students into the pipeline for STEM fields.
"To be sane in an insane world is insane." Maybe we can only fight bias with other biases? Maybe "If you can't beat 'em, join 'em"? Sep 20, 2022 at 15:42
Maths is good and all, but it's important to consider The Unreasonable Ineffectiveness of Mathematics in most sciences
For me, Plato has a lot to answer for. As I see it, he took the disruptive cult of Pythagoras, and the iconoclastic discourse of Socrates, and fused them into something workably self-perpetuating - the Academy. But while Tegmark and others still get misty-eyed over triangles, no one takes Plato's views about the solar system relating to Platonic Solids and musical notes seriously. Sabine Hossenfelder goes into this area in her book Lost In Math, and her talk How Beauty Leads Physics Astray. At the same time Dmitri Mendeleev was inspired by musical keys to create his periodic table, and there is a deep connection in terms of waves there, even though it is nothing like he imagined.
As I see it, truly universal math relates to what is most intersubjective. But we have be careful to distinguish pre-conceptual considerations and deriving logical forms, which is a domain for philosophy, and analytic which is the domain of mathematics where logical form and definitions have already been arranged - see Are the concept of time and space apriori to natural language or are they just references within natural language? and Relationship between early Wittgenstein and late Wittgenstein
Obviously I don't want to say 2+2=/=4 because it was old white guys decided that. But, there is a process of going from experience, to abstraction, and we have to be wary about what we draw from abstraction that has social consequences, including hidden or implicit ones.
So consider beauty. It is just a fact that we find more symmetrical faces more beautiful, which has evolutionary roots. In physics, Noether's Theorem shows us that conservation laws are literally just another way of saying, there are symmetries under specific transformations. So, are conservation laws, beautiful?
"There is no excellent beauty, that hath not some strangeness in the proportion." -Francis Bacon, in Of Beauty
There is a big difference between mathematics, and the inferences we draw from it's operations. In reflecting on time, we find our intuitions highly suspect and obstructive, and the nature of it very much not resolved (it has no explanation in Quantum Field Theory, and Relativity where if is a dimension appears irreconcilable with that seemingly deeper picture). We feel like 'beauty' is this objective thing - like Keats blithely and wilfully lied about it: " 'Beauty is truth, truth beauty', --that is all Ye know on earth, and all ye need to know". While it can help in what Popper drew attention to as a not-algorithmic process of hypothesis generation, it's a terrible guide for picking between theories.
It'd be like insisting a spinning compass needle shows us what North is, instead of understanding that compasses have been useful in finding North. Mathematics is a system of abstractions, that we endeavor through learned practices to make as intersubjective as possible, but we are humans, trapped in our historical social moments. Axioms turned out not to be 'self evident truths' as Euclid thought, but instead sets of choices.
Mathematics is a set of tools we use, like language, and while we have established formalisms it is also a domain of creativity. It can be hard in teaching to get children to understand that, and indeed many adults don't, even some mathematicians. We know there are links between thrown-object play and certain mathematical competencies, and links between musical and mathematical competencies. I'd relate these especially to earlier stages of learning, and getting to the point of seeing how there can be fun, within a framework. A great mathematician like a great poet, takes well-worn parts and finds new ways to relate them - I always think of Ramanujan, and Godel, as singular characters finding their own expressions.
Can mathematics be racist is not so different from, can language be racist. It is if you do racist things with it, and that doesn't always mean knowing you are. The legacy of math-mysticism still around in Mathematical Platonism seeks to put math on a special pedestal, and while properly used it accesses our most intersubjective knowledge, it does not access some noumena, and we should be more wary than Plato of believing the cosmos is a vast pipe-organ because that makes us feel good - or that there is only one kind of smart and it relates to a narrow picture of rote math competence. Our real intelligence is about connecting to our world as a whole (see Do IQ tests measure intelligence?), and great mathematics is not just about working with abstractions but questioning what the useful abstractions are.
2I would call it Engineering to put abstractions to use. Much of it is simplifying, making approximations and disregarding non-essentials. "Improvement makes straight roads, but the crooked roads, without improvements, are roads of genius." My mother would say, we need fewer Chiefs and more Indians. Well, a long time ago she did. Sep 20, 2022 at 17:00
@ScottRowe: But if we call math 'play's, or fundamental research or whatever, what kinds of things we choose to work on & develop are still crucial, & who is allowed to speak. Imagine if Ramanujan had got better support. Or Galois hadn't died, probably linked to having too-radical politics Sep 20, 2022 at 19:05
1Politics: poly meaning 'many', ticks meaning tiny blood-sucking insects. Sep 20, 2022 at 19:31
One of the ambiguities that plagues the philosophy of science is whether “science” refers to the set of universal truths (including as a subset forensic truths) that scientists aim at discovering, or whether it refers to the activities of scientists. Similarly, it's not clear from your question whether “mathematics” refers to the truths that mathematicians strive to discover, or whether it refers to the activities of mathematicians. (We might further narrow that to “the collective activities of mathematicians” — that would have implications for the questions.)
It's very difficult to imagine how the Pythagorean theorem is racist, but very easy to imagine that the activities of mathematicians could be racist — however that term is construed (whether individual or collective, intentional or unintentional, etc.).
That's a strange dichotomy to start with, and also a very Plantonist view of "science" which is not agreed upon by even a simple majority of philosophers.– BurnsBASep 20, 2022 at 15:14
I think about this often. Not every noun can be verbed, or should be. Sep 20, 2022 at 15:39
Let me try to argue that Mathematics is culturally biased.
The mathematical logic derives fundamentally from the following proposition, which I am arguing is culturally biased:
"A thing cannot be A and at the same time not A."
It is called "Law of Noncontradiction". Some oriental culture/or philosophy rather promotes that "A" and "not A" at the same time is rather the true aspect of the reality. An example would be "Yin/Yang" philosophy. In Korean culture heavily influenced by such thought tradition, we rather view things not in dichotomic way but as in more holistic way.
6Yin and yang is a concept that describes opposite forces which are interconnected. I very much doubt that anyone who holds to this philosophy would say that "A" and "not A" can both be true at the same time in a logical sense. Sep 19, 2022 at 10:12
1@NotThatGuy You may be interested in constructive mathematics, or alternative foundational theories for doing mathematics, where the: "law of the excluded middle is removed" and whilst "true" and "false" are concepts, "not true" does not imply "false" (or something like that,... I don't really understand it as I, as many mathematicians/students, steer well clear of foundations)– FShrikeSep 19, 2022 at 10:20
3You might be closer to a valid argument if you suggested Schrodinger's Cat as being <A & ! A> rather than misinterpreting ancient Asian (not "oriental") cultures. Sep 19, 2022 at 12:01
3What you describe is the "law of excluded middle" and is one axiom in classical logic, but not intuitionist logic. I also don't think Yin/Yang refers to truth values in logic.– qwrSep 19, 2022 at 23:56
The Taiji is said to have started from observing sunny & shady side of a mountain, & is often described as two fish yoked together. Daoist & Buddhist thought see limits on the power of conceptual thought, which seems more relevant. See 'Philosophers or philosophical traditions that reject symbolic reasoning' philosophy.stackexchange.com/questions/82360/… Buddhists are strongly associated with acceptance of zero bbc.co.uk/programmes/p0038xb0 While Daoist alchemy changed the world (gunpowder) Sep 20, 2022 at 19:47
Mathematics in and of itself is, in my opinion, completely neutral. But what can be descripted or in any other way done with mathematics can be highly discriminatory and much more and much less.
That said and looking at some other answers here, of course there can (,as always: sadly) be discrimination and/or different things throughout the creation of sth, but that does not make the thing itself discriminative or else. And if you want to go so far as to say that it could be seen as, i will no simply switch to short words that i think can be assotiated with it t shorten this text, bad because it uses letters created by just one specific region in the world, then again that does not change the fact that math in it‘s core is logic and this logic …..
P.S.: I accidently stumbled upon this site and question and while I find myself getting quite philosophical a lot of times, these philosophic thoughts on this site seem to get a bit too far for my taste. Of course that does not mean anyone can‘t do it - just my opinion. Of course everything can be misunderstood. Of course you can philosophize about a whole lot. But at some point you should ask youself what point this specific philosophizing has in some cases. If you at some point discuss wether being is being then.. . Also: at extend this ‚philosophy‘ is just discussion and opinions. And if it‘s just a joke for you than I find it misplaced here anyways. Anyways i could say so much more about this but this is already not part of the actual answer so i will just leave it at this.
In itself Mathematics is more or less neutral in the modern day but it must also be realized that any human act can only happen in a context. So, it consists of vastly different things to study Mathematics in a developed vs underdeveloped one one.
For example, it's very difficult for a person to consider any other career paths except one which rise them on the socio economic ladder when they come from an underdeveloped place. As a personal example, in India, many people aspire to become engineer or doctor just because there is not much economic prospect for mathematics research or so.
Another point is that of social discrimination. I believe even in America there is a showing of less number of certain ethnicity group in higher stem fields. The explanation given is the societal problems bled into the college admissions and such
I think it is also sensible to believe that there is a cultural aspect to it as well because if we compare across many different groups in a fixed time with similar conditions, some subcultures are much more scientifically productive than others.
One big example for me is Jews jn the 19th century. Ever single name related to Physics which come in average HS textbook in this century seems to be a jew. Take for example Richard Feynman, Albert einstein, Bohr etc.
3"Another point is that of social discrimination. I believe even in America there is a showing of less number of certain ethnicity group in higher stem fields." Unequal outcomes do not imply discrimination. Sep 18, 2022 at 20:30
1See here . under represented mathematicians Sep 18, 2022 at 21:12
5Unequal outcomes do not imply discrimination. For example, see: sciencedirect.com/science/article/abs/pii/S0163638300000321 nature.com/articles/srep19669 onlinelibrary.wiley.com/doi/abs/10.1002/icd.2064 sciencedirect.com/science/article/pii/S0960982210014491 link.springer.com/article/10.1023/B:SERS.0000011068.42663.ce onlinelibrary.wiley.com/doi/abs/10.1111/… pubmed.ncbi.nlm.nih.gov/19883140 psyarxiv.com/zhvre Sep 18, 2022 at 21:21
2I believe it's a matrix of issues. The present discrimination is most likely due to a historical reasons. Now, people have forgotten those reasons but the negative effects on the minority group still exist. As in, due to discrimination in the past, their identity group could not amass much social wealth and hence now they're disadvantaged. I think there is an ocean of difference if one has a family who were well of for ten generation vs only one or two. Sep 18, 2022 at 22:00
2@TrystwithFreedom There is certainly discrimination in favor of some ethnic groups that are deemed "underrepresented", and against others that are deemed "overrepresented", as evinced by affirmative action policies. Sep 18, 2022 at 22:03
Mathematics is founded on shared culture.
The language of mathematics is a product of shared culture, and the work done by mathematicians is subject to the internal politics of academic institutions and economic forces. There are multiple foundations of mathematics; choice of foundation is guided by philosophy of mathematics. Professional mathematicians work primarily using intuition, and communicate in a way which is highly dependent on shared background.
There is no single foundation of all mathematics.
There are multiple foundations of mathematics, which:
- do not necessarily ask the same questions,
- may give contradictory answers when the questions are the same, and
- demonstrate answers in different ways.
Although foundations themselves are mostly formal, the choice of foundation is philosophical and practical in nature. Either there is no one true foundation of mathematics, or we have no objective way to determine what it is. There is not any single way to choose a foundational system, for example:
- a realist will choose axioms which they believe are true,
- an intuitionist will choose axioms which reflect their understanding of mathematics,
- a constructivist will choose a foundation in which proofs construct (or perhaps even compute) examples, and
- a pluralist will use multiple foundations to explore a subject.
These sorts of philosophies give rise to many foundations, for example ZF, ETCS, or HoTT, plus your choice of axioms.
Historically, philosophy of mathematics was even more strongly dependent on philosophy and religion in general; refer to e.g. Pythagoras, Cantor.
Mathematicians operate informally, based on a shared background.
ZFC is the most popular foundation because of its presence during the foundational crisis, and remains popular due to momentum. Saved from crisis (for now), there hasn't been much incentive to use different foundations; most professional mathematicians don't care about foundations at all, as long as there is one:
- they don't need their proofs to compute,
- they don't need their proofs to be "portable" to other categories,
- they rarely need to formalize, and
- they have no problem using "abuse of language" to get the niceties a different system would provide.
But the indifference of professional mathematicians to foundations is more evidence that mathematics is cultural. Anyone who accepts a particular foundation will be able to verify a fully formal proof and accept its result (at least up to issues like the original four-color theorem proof, which was computer-generated and too large to verify by hand). However, most mathematical papers aren't formalized; most proofs are written as sketches of reasoning which represent the bare minimum to be understood by other experts in the field, not by novices or formal verifiers. A mathematician who had not been taught these informalities would not be able to fill in the leaps of reasoning made when reading an actual mathematical paper. Even an expert mathematician will not be able to read a paper in another field without learning the background and conventions of that field first.
In mathematical practice, the purpose of a proof is not to formally verify a claim, but rather as a tool to convince your peers that your claim is correct relative to your shared intuition. Dictionaries and grammar books describe a common subset of language that most speakers accept, but can never hope to capture the full meaning of the words people speak, which is dependent on shared context; foundations of mathematics underlie the language of mathematicians as a common subset that most mathematicians accept, but neither describes the intuitive reasoning nor communication of mathematicians. A native speaker doesn't care what dictionary you're using because you know what that word means.
The language of mathematics is developed informally and collaboratively and based on shared intuition. Not only is there no objective choice of foundation, there is no single foundation underlying mathematical practice; to verify the proof, you must understand the field.
It's not just about knowing enough.
Sometimes one way of looking at something is more intuitive, or sometimes a different perspective helps. The way in which mathematics is presented will prioritize different problems and make some results easier to find than others.
Different mathematicians' minds work in different ways. This is partially the individual's innate traits, partially their background, and partially their approach to the particular problem at hand.
Avi C's answer goes into more depth about how presentation matters. I'd like to emphasize that choice of abstraction also really, really matters. Very different-looking mathematical structures can turn out to be equivalent, and which one you're studying can make a big difference; it can be like the blind men and the elephant.
Mathematicians develop their skills and background, and choose their research, dependent on external forces. These may be characterized by the internal politics of academic institutions and economic forces. Some factors which influence mathematicians include:
- the background they already have and education they received,
- what topics are popular or important at the time,
- the opinions of their advisors and the esteem of their peers,
- funding, and
Mathematics is not only not acultural, but essentially is a culture unto itself, with:
- underlying philosophy and belief systems;
- a complex, subfield-specialized, and highly-developed language;
- a strong dependence on individual background and approach; and
- institutions which are influenced by both internal and external political forces.
Mathematics is not culturally neutral, because of
Base 10 counting*
We are all familiar with the positional number system where each position represents a power of 10. It is a good system for the purpose - clear and readable. We could also use roman numerals - they are not as convenient for calculations, but still usable to average person. This system is also based on number 10 - X=10, C=100, M=1000 and so on.
However, there are other number systems possible.
Imagine that you can only use two digits, one and zero, to represent values in positional manner. This is binary.
We only started using binary, or base2 recently, with introduction of computers, due to physical way the digital logic circuits work.
Around the same time also started using octal (base8) and hexadecimal (base16), because that allowed for more concise representation of binary values.
Here raises the question - why 10?
Why not 8, a convenient number for computer sciences, or 12, that has many divisors?
The answer is obviously "because we have 10 fingers".
But if we were visited by aliens, they would be puzzled why 10 - a number with only 2 divisors, and one of them is an odd prime(!).
While base of the numeral system does not change anything in how mathematics work, it does influence the areas we develop.
Take the division rules - if sum of digits is divisible by 3, the number is divisible by 3 and so on. This works for decimal system, but does it hold for octal? Hexadecimal? Would "aliens" using different base even find such rules?
[Note: I actually don't know, but it seems the rules are different for different bases]
It also influences how we write "decimal" fractions.
1/3 in decimal = 0.33333...
1/3 in ternary (base3) = 0.1
I would like to note that learning to do basic arithmetics in base other than 10 is both enlightening and mindboggling (when fractions are involved)
So in the end, yes, math is culturally biased.
*This answer is in base 10. Since that could be misinterpreted, 10 is defined as this number of sticks: ||||| |||||
No mathematics is not politically neutral in todays highly technological society. A case in point is the Black-Scholes model in mathematical modelling of the financial markets with financial derivatives.
In books I looked at it was often referred to as calculating the "fair price". As we discovered durimg the financial debacle of 2008 that nearly took down the global markets down they were doing little of the kind - instead they were spreading the contagion around. There is after all, something called "moral hazard" and the macro & micropolitics of large investment firms and banks that are rather hard to price in.
It was probably politically neutral during the long learning period whilst we began to umderstand what the physical and mathematical sciences could do - but no longer.
Also, I would dispute your characterisation of mathematics as the invention of "dead white greek males". Plato himself reprimanded the Greeks for thinkimg every invention was their own. He pointed to the Egyptians. Now Kronecker said, "God created the integers and the rest is the work of man" pointimg to the primacy of the integers. But the oldest evidence we have for numbers are the Ishango bone, a baboon fibula found in the Congo. So it was definitely, a black African man in Africa who invented/discovered numbers and who might have been inspired by God - but definitely he was looking at the stars.
2Except he wasn't black. No one was black then. Sep 20, 2022 at 14:32
"the oldest evidence we have for numbers are the Ishango bone" The Lebombo bone is older. And you cannot conclude that either was "definitely" the first. Sep 28, 2022 at 17:57
There are well-known examples of communities which have written into law that pi shall equal three, or similar assertions. These sometimes come from taking religious sources over-literally, but there's no reason something of this sort couldn't be politically motivated.
In those cases, where the culture or subculture asserts something that is demonstrably false, science and mathematics are unavoidably going to conflict with the local culture. And folks defending those assertions are going to advance arguments for why any evidence or proof is invalid, by impugning the source or the methodology or simply declaring it Not What We Believe. This may not be accessible to rational discussion; the best you can do may be to try to understand where they're getting this belief from.
That may or may not answer your question.
And I'm going to stop there before I get political.
1"taking religious sources over-literally" Or overly-abstractly. It's very possible for a bowl to have radius 1.00 X and circumference 3.00 X, if we allow the circumference to be measured at the inside of a lip with non-zero thickness. Or allow the bowl to be either less than or more than exactly half a sphere. Only when the bowl is imagined as an abstract zero-thickness semi-sphere is there a problem. Sep 20, 2022 at 21:29
People discussing mathematics who are not reachable rationally have some kind of psychiatric problem. It is a contradiction in terms. Sep 24, 2022 at 3:38
Scott, if only that were true. .– keshlamSep 24, 2022 at 3:48
No it's not....
Okay, this answer is based on a book called "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World".
There were some members from Society of Jesus who were like some sort of Social engineers in the 16th century and they were in charge of what ideas would be allowed to continue to run into society. They tried to paint the idea of infinitesimal as a total nonsense.
For, strange as it might seem to us, the condemnation of indivisibles in 1632 was not an isolated incident in the chronicles of the Jesuit Revisors, but merely a single volley in an ongoing campaign. In fact, the records of the meetings of the Revisors, which are kept to this day in the Society’s archives in the Vatican, reveal that the structure of the continuum was one of the main and most persistent of this body’s concerns. The matter had first come up in 1606, just a few years after General Acquaviva created the office, when an early generation of Revisors was asked to weigh in on the question of whether “the continuum is composed of a finite number of indivisibles.” The same question, with slight variations, was proposed again two years later, and then again in 1613 and 1615. Each and every time, the Revisors rejected the doctrine unequivocally, declaring it to be “false and erroneous in philosophy … which all agree must not be taught.”
(Chap-4)Tacquet’s claim to mathematical fame rested chiefly on his 1651 book Cylindricorum et annularium libri IV (“Four Books on Cylinders and Rings”), in which he showed a complete mastery of the full mathematical arsenal available in his day. He calculated the areas and volumes of geometrical figures using both classical approaches and the new methods developed by his contemporaries and immediate predecessors. But when it came to indivisibles, the usually mild-mannered Jesuit turned blunt:
I cannot consider the method of proof by indivisibles as either legitimate or geometrical … many geometers agree that a line is generated by the movement of a point, a surface by a moving line, a solid by a surface. But it is one thing to say that a quantity is generated from the movement of an indivisible, a very different thing to say that it is composed of indivisibles. The truth of the first is altogether established; the other makes war upon geometry to such an extent, that if it is not to destroy it, it must itself be destroyed.
Destroy or be destroyed—such were the stakes when it came to infinitesimals, according to Tacquet. Strong words indeed, but to the Fleming’s contemporaries, they were not particularly surprising. Tacquet was, after all, a Jesuit, and the Jesuits were then engaged in a sustained and uncompromising campaign to accomplish precisely what Tacquet was advocating: to eliminate the doctrine that the continuum is composed of indivisibles from the face of the earth. Should indivisibles prevail, they feared, the casualty would be not just mathematics, but the ideal that animated the entire Jesuit enterprise.
Tl;dr: Calculus... Christianity.... and Society of Jesus?!?!
Here is another MSE post discussing the same
I read this book in one of the joyful times of my life. It was in the Thiruvanthapuram State Library. There was a nice park nearby and I laid on the grass reading it for a few days before I gave up lol. Met some nice people there. Sep 20, 2022 at 22:29