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One could argue that, fundamentally, mathematics is a sociological process, as the backbone of mathematics is that of the mathematic proof, and the mathematic proof of a statement, at least as used in common day, is a method of communicating that the statement's truth to another individual.

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Eugenia Cheng, Morality in Math

In the diagram, the two individuals understands the proof by different reasons, however they can both agree on the truth of the statement virtue of it's proof.

What are some examples of other sociological considerations in mathematician culture? For instance, research culture?

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fundamentally, mathematics is a sociological process

Could you argue that engineering is 'fundamentally' sociological? Either a bridge stands, or it falls. Either an engine works efficiently, or not. And, the principles determining that, depend on accurately and consistently quantifying the systems involved mathematically, to draw the correct inferences.

I argue in this answer, that the utility of mathematics derives from it's intersubjectivity, which relates to the fact we all have to be experiencing the same number of dimensions, plus conservation laws (which are continuous symmetries under transformation): The Unreasonable Ineffectiveness of Mathematics in most sciences Many things could be different in another person's experience, but if they had less or more dimensions, their chemistry and biology simply would not work.

Science is the process of developing tools to try and sift out intersubjective data, what would be the same for different observers etc, from subjective reactions which may result from say cognitive biases grounded in our personal life history. We can study science sociologically, but to argue it is fundamentally sociological, as Kuhn did, is problematic to an unsustainable degree. Discussed here: Why is postmodernism apparently so ill-perceived in philosophy of science?

However, it is interesting to look at mathematics culture in relation to paradigm shifts, like the arrival of non-Euclidean geometry, or imaginary numbers. Rather than sweeping previous work away, these seem more to me like a hard-fork in the development of a blockchain ledger, or the branching off and divergence of a linguistic or evolutionary subgroup.

There definitely are approaches to applying more sociology to teaching math, like the Equitable Math Foundation. How constructive their work is, remains a subject of controversy.

There's quite a bit of research on the correlation between success in mathematics, and in sport, especially during early teen education. Eg The Association Between Physical Activity and Mathematical Achievement, Being good at math might help you become great at sports. This is interesting in terms of class and gender impacts at ages where confidence in subjects is shaped.

Mathematics is a language, and pushing into new areas with it involves substantial creativity, which should be more widely known. It's fascinating to consider the quirks of unusual major figures like Ramanujan, or Godel. Issues of neurodiversity-acceptance have a special bearing on the work of academic mathematicians and theoretical physicists.

Emmy Noether could perhaps have achieved a lot more, but for gender bias, and she should be much more widely celebrated. Who and how we remember people is important to future mathematics and society.

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    whether a bridge stands or it falls is objective. The German government is about to bulldoze an entire street of popular nightclubs to build a highway. Is that objective? What if their calculation says the financial value of the street is lower than the financial value of the highway - is it objective then?
    – user253751
    May 16, 2023 at 15:04
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There's a lot of fallacious concept's in the article, and the OP. It's necessary to have them clear for a proper answer.

While Logic is the FORMAL set of rules that govern reason, Mathematics is a type of knowledge consisting of an abstract description of the world. As such, Mathematics is an individual (rather than social) asset. This is how it develops:

Following Kant, you need to know the form of an apple -a sphere-, in order to know an actual apple and vice-versa: you need to know an apple in order to know the sphere. This is essentially how a priori knowledge (Mathematics) is developed: simultaneously to a posteriori knowledge (empirical knowledge, objects of the world). The apple is a PHYSICAL object, and the sphere is a METAPHYSICAL object that correspondiente to it.

Please remark that in the Kantian jargon, a priori does not mean "before", but instead "necessary for". The key contribution of Kant lies here: metaphysical truth (e.g. Mathematics) is possible, exists, and it develops simultaneously to empirical truth (e.g. physical objects).

From the moment you know the object, you start to develop your own mathematical knowledge: more than one apple is two apples (addition), dividing such system provides two groups of a single apple (division, subtraction, etc.). Check Kant's discussion about the Transcendental Analytic a priori knowledge: Kant's categories.

Mathematical truth, in this context, is just the logical CONSISTENCY between the facts of the world (apples, as perceived by our senses) and mathematical knowledge (spheres, Kantian categories).

Now, mathematics has the following functional structure:

  • Mathematics is a TOOL that helps knowing the world and predicting the future (what has been described, for example, if I go get an apple, I can predict that I will have more than what I have now, and if I have 1, I can add what comes: 1+1=2).
  • Mathematics is a LANGUAGE, which allows communicating mathematical ideas. Here, it becomes SOCIAL. In this context, mathematical truth is not only CONSISTENCY, but also AGREEMENT. I use the word "seven" or the symbol "7" to describe something, that MUST have the same meaning for the rest of my social group (agreement). In other words, mathematics, as a LANGUAGE is an OBJECTIVE (not only SUBJECTIVE) agreement about its CONTENTS (objective essentially means shared subjectivity). So, mathematical truth is the shared-subjective (OBJECTIVE) agreement about the CONSISTENCY between the PHYSICAL facts of the world, their METAPHYSICAL representation (Mathematical)

Now, if you want to address the Mathematics social process (the social process of the development of Mathematics), the previous concepts must be crystal clear. Eugenia Cheng's article, -in form and content- is poor in concepts. Perhaps it is Ok for a TV audience, but not to make philosophy,

MORALS is a set of RULES that help social coexistence. For example, if the final, shared goal of the group is to exist, and grow, instead of dissipating, Morals is the set of rules that determine how to socially interact in order to survive.

Chang's definition of mathematical morality is this:

Mathematical morality is about how mathematics should behave, not just that this is right, this is wrong.

If so, Chang should have used "mathematical behaviorism" (behaviorism is understood as "way of thinking" when its subject is passive, unable to behave).

Well, Mathematics don't "behave", because TOOLS or LANGUAGES (which are ultimately communication TOOLS) don't behave. Tools are just resources for survival.

In any case, Chang's poor definition of mathematical morality is derived from social morality, regarding an absolute imperative "how X should behave", which has no sense. Morals are subjective, RELATIVE to each society, they are not ABSOLUTE, for all humanity; as said, the moral rules of group A, which has the goal of survival, might be radically different than those of group B, which has the goal of making war. Moral rules are relative. So, no mathematical morality.

There might be few or none sociological concerns regarding Mathematics, simply because, as said, Mathematics is a TOOL, and tools are far from presenting sociological considerations (commonly, guns are considered a social problem, but in final terms, the concern is never about guns; guns as such are just extensions of our fingers, if you stick a needle to your hand, you need to avoid scratching your eyes, the problem is not needles). There are no sociological considerations regarding tools (what are the sociological considerations of a Phillips screwdriver? What are the sociological considerations of integer addition?)

On the contrary, there might be sociological considerations related to the risks or impact of disciplines sustained by Mathematics, but this the same as the risks or impact of discipline X. Mathematics is not part of the problem. For example, https://en.m.wikipedia.org/wiki/Social_constructionism, which might raise the conflicts between social goals and the origin and application of the tool.

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  • "consisting of an abstract description of the world" Surely, mathematics is not tied to the world, eg geometry in very high dimensions. You seem to have in mind physics..? "Mathematics is an individual asset" What on Earth does that mean? "the form of an apple -a sphere-" ?? "There are no sociological considerations regarding tools" What about eg nuclear weapons..?
    – CriglCragl
    Apr 20, 2023 at 12:24
  • @CriglCragl A) Mathematically, one [1] ABSTRACT object can be 1x; objects addition can be 1x+1x=2x, that is not physics, physics is about more CONCRETE, physical objects (things). B) Misquoted statement. C) Apples are not spheres, and no object in the whole universe is a (perfect) sphere; ergo, an apple or a planet can be seen as spheres: tolerance is subjective. D) There are no sociological considerations regarding tools: we address the core goal (killing people), not one of its components (using copper cables / using nuclear weapons / using the thumb or the pinky / using Windows 95...).
    – RodolfoAP
    Apr 20, 2023 at 13:20
  • Mathematics may have begun for us from the world, but it can go far beyond ties to it. What did you mean by what you said though, why the word 'personal'? We don't get sphericalness from apples, you have already to be in abstract to concieve them. But the fact a certain assembly can do a certain thing, creates it's own special dilemmas, surely? My views on math here 'The Unreasonable Ineffectiveness of Mathematics in most sciences' philosophy.stackexchange.com/questions/92058/…
    – CriglCragl
    Apr 20, 2023 at 13:34
  • "Mathematical truth, in this context, is just the logical CONSISTENCY between the facts of the world (apples, as perceived by our senses) and mathematical knowledge (spheres, Kantian categories)." Can you explain what fact of the world corresponds to the snake lemma in homological algebra? en.wikipedia.org/wiki/Snake_lemma
    – Frank
    Apr 20, 2023 at 14:07
  • @Frank Make it simpler, ask yourself "what fact of the world corresponds to 10¹¹¹¹ ?": no such fact exists. The statement doesn't imply each mathematical proposition corresponds to a world fact. See Formal System.
    – RodolfoAP
    Apr 20, 2023 at 14:44

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