To "enrich" as opposed to "dissolve" the point?
Grothendieck dissolved the classical notion of a point into a functor of points$$h_X : (\mathbf{Sch})^{\mathrm{op}} \longrightarrow \mathbf{Set}, \qquad h_X(S) = \mathrm{Hom}(S, X),$$re-imagining...
View ArticleIs Calculus Abstract Newtonian Mechanics? [closed]
If I claim that Calculus is just an abstraction of "Newtonian Mechanics" , would you agree with me? If yes or no, provide me with the reasons for the same.
View ArticleFirst-order logic without equality
Can we do without equality in first order logic? I looked at some cases in which equality is essential and found that it seems enough to have inequality implicit in the variables. Let $\phi(x,y)$ be a...
View ArticleAre there other approaches for the foundations of mathematics, other than...
Are there other approaches for the foundations of mathematics, other than logic and set theory?And why does set theory begin talking about objects and groups of objects.Is it proven somewhere that that...
View ArticleCan Aristotle’s notions of potency and act be applied to mathematics? [closed]
Can Aristotle’s notions of potency and act be applied to mathematics?I ask because I think it can for consider the following. Suppose we had a circle. Then that circle is potentially a line because I...
View ArticleWhat is the argument supporting the "monist" or "universist" conception of...
Preemptive Prefatory Note: This is not an "opinion-based" question in the sense that I am not asking which philosophy of set theory is the "right" one. Rather, I am seeking to know what exactly is the...
View ArticleIdentity of indiscernibles and the Axiom of Choice [closed]
I read Max Black's argument against the identity of indiscernibles his 1952 paper "The Identity of Indiscernibles" and I'm now doubting the axiom of choice for finite sets.His argument basically goes...
View ArticleWhy linear congruential generator is called random number generator?
As far my understanding linear congruential generator is used to produce pseudo random number. But the algorithm requires four parameters like,def lcg(modulus, a, c, seed): while True: seed = (a * seed...
View ArticleHow do mathematics define a point?
I have a serious doubt. How do mathematicians define a 'point' in a space or a plot? If we have a clear explanation for a 'point' , I think my doubt on infinitesimals and infinity will be clarified.
View ArticleI'm having confusion with how Albert Meyer defines an axiom.
In the book "Mathematics for Computer Science", Meyer refers to an axiom as a "proposition that is accepted as true." To give some context, he explains this with one of Euclid's postulates which states...
View ArticleWhy are sizes of infinite sets equal if theres a bijection between them?...
We think of finite sets A and B having the same size whenever we can draw a one-to-one correspondence between the elements in A and B, and vice versa, which is based on our intuition for finite sets....
View ArticleExactly who popularized the modern definition of domain and codomain of...
In Whitehead and Russell's Principia, domain is the referents of relation; converse domain is the relata. Modern function in mathematics is just one special case of relation whose referent is unique...
View ArticleIs there an analogy between category errors in philosophy, and type errors in...
In philosophy, a category error is predicating of something a predicate which it does not even apply to. For example, it is a category error to ask whether or not the number $4$ is happy. I have been...
View ArticleDoes this article misapply a result in mathematical logic?
A recent neuroscience paper (Milinkovic & Aru 2026, Neuroscience and Biobehavioral Reviews) argues that biological computation differs fundamentally from digital computation, partly on the basis of...
View ArticleWhat is a number?
A dictionary I consulted said a 'number' is a 'quantity', so I looked up what quantity means and the same dictionary said it is an amount or number of some material or thing. Since quantity and number...
View ArticleDoes every computable ordinal have a "true" fundamental sequence?
A fundamental sequence of a limit ordinal is usually defined as a strictly increasing cofinal sequence of ordinals all below a given limit ordinal. One application of this is in the fast-growing...
View ArticleGeneralizing Bertrand Russell's "Pope" proof: Can any false premise $P$ lead...
Recently, I came across a famous anecdote regarding Bertrand Russell and the principle of Ex Falso Sequitur Quodlibet. As the story goes, a student challenged him: "If $0=1$, prove that you are the...
View ArticleWhat is it we are actually doing when we assume a statement $A$ for the sake...
A very simple question has been bugging me for a very long time now: what is it we are actually assuming when we assume $A$ during natural deduction proofs? (i.e. write "Suppose $A$...")Or 'What are...
View ArticleWhat is the role of mathematical intuition and common sense in questions of...
I got the number$$\frac{\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)}{\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\right)}=0.824326275998351470388591998726842...$$in the...
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Why do circles underlie the measurement of distance and translation? [closed]
In Euclidean geometry and, more generally, in metric spaces, distance is tightly bound to circles (or spheres): a circle is defined as the set of all points at a fixed distance from a center, and...
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