2
votes
1answer
99 views

Why didn't Frege succeed in his attempts to reduce mathematics to logic?

My background: Sophomore-level understanding of mathematics and philosophical logic. All the explanations I have found online so far are either far too technical or too simplistic. Thanks in advance ...
6
votes
6answers
212 views

Interviews of famous modern mathematicians

I was wondering, are there any good collections of interviews of famous modern mathematicians? It can be text interviews, or audio or video recordings. I am not sure what exactly I mean by "modern". ...
3
votes
1answer
161 views

Errors of Euler interpretation?

To complement the recent post on Euler's errors, I would pose the following question: what common errors of Euler interpetation appear in the literature? What errors are attributed to Euler's work in ...
2
votes
1answer
55 views

Need help locating a paper

One of the references of the paper Paulo Régis C. Ruffino, A Criticism on "A Mathematician's Apology" by G. H. Hardy (arXiv:1112.4499 [math.HO]) is: Vershik, A. M. – A Dangerous Joke, The ...
3
votes
4answers
142 views

Soft question: Examples where implications derived from mathematical models failed to describe reality

I have always been fascinated by how well conclusions drawn from mathematical models could fit reality, so I wondered if there are any counter examples. In "Gödel, Escher, Bach" I could already find ...
7
votes
2answers
226 views

How to explain ✳43.3 and ✳43.31 in Whitehead and Russell's PM?

Take ✳43.3 for example, I presume $ P = R |Q $ where R is fixed. $ R| $ is the relation between $R|Q$ and $Q$, ie. $ R| = \hat{P} \hat{Q} \{ P = R|Q \} $ $Ɑ‘R|= \hat{Q}\{ E! R|‘Q \}$ Given that ...
7
votes
1answer
263 views

Exactly who popularized the modern definition of domain and codomain of functions?

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 ...
1
vote
3answers
136 views

When and where the concept of valid logic formula was defined?

I was stimulated by a recent question about Gödel Completeness Theorem. All my citations are from Jean van Heijenoort (editor) From Frege to Gödel A Source Book in Mathematical Logic (1967). Gödel's ...
2
votes
1answer
99 views

Why can't ✳1.1 be expressed symbollically in Whitehead and Russell's PM?

✳1.1. Anything implied by a true elementary proposition is true. Pp. In the follow passage, it says, "we cannot express the principle symbolically, partly because any symbolism in which p is ...
0
votes
3answers
62 views

Should the notion of continuity, usually ascribed to Cauchy, be ascribed to Leibniz?

In his text, Deleuze and the History of Mathematics, Simon Duffy writes: Leibniz also thought the following to be a requirement to continuity: "When the difference between two instances in a ...
4
votes
3answers
312 views

What does “meaning” mean in Whitehead and Russell's PM?

In Principia Mathematica's Introduction, there is a definition for "incomplete" symbol: By an "incomplete" symbol we mean a symbol which is not supposed to have any meaning in isolation, but is ...
1
vote
2answers
124 views

What is the necessary condition for the process of “proceeding to the limit” in Whitehead and Russell's PM?

I read this from Introduction of the 1st edition of Principia Mathematica by Whitehead and Russell: Since the orders of functions are only defined step by step, there can be no process of ...
0
votes
1answer
37 views

Math Mindeset: Historical Learning vs Generality of Concepts

I started math four months ago with modules like measure theory and topology. It was unavoidable to notice how many concepts are more general than what I thought before. For example the ...
4
votes
4answers
208 views

What is the difference between asserting “$\phi(a)$” and asserting “$\phi(a)$ is true” in Whitehead and Russell's PM?

The first edition of Principia Mathematica clearly distinguishes "Socrates is a man" and "'Socrates is a man' is true." Judging from the context, the distinction is neither a primitive idea nor a ...
6
votes
1answer
171 views

How does Schröder explain the apparent oddity of ❋5.11.12.13.14 in 1st ed of Whitehead and Russell's PM?

The footnote refers to Schröder's work. I'd appreciate if someone can explain Schroder's insights and spare me some hard reading.
4
votes
2answers
186 views

How to prove ❋4.86 in 1st ed of Whitehead and Russell's PM?

This one has a great degree of self-evidence. Paradoxically, I find it difficult to deduce it from primitive propositions. The book only hinted ❋4.21 and ❋4.22.
3
votes
0answers
68 views

Is there a link between level of abstraction and use of numbers?

One of my friend who stopped studying maths in high school told me once You study maths, can you help me fill my tax forms? In her mind, advancing in maths studies implied manipulating an ...
5
votes
1answer
109 views

Does Whitehead and Russells' PM distinguish Proof from Demonstration?

I'm currently at Chapter 4, vol. 1 and 1st ed. I have to ask this question because the most important thing about this book is in its minute details. Thanks. Take *3.3 for example. Acording to this ...
5
votes
1answer
77 views

What is $M_x$ in Frege's Basic Law IIb?

Gottlob Frege's magnum opus, "The Basic Laws of Arithmetic" (Die Grundgesetze der Arithmetic in German) constitutes one of most impressive and meticulous attempts at developing a rigorous foundation ...
0
votes
2answers
109 views

General questions about theorems and laws

I have doubts about the construction of mathematical elements. There are proofs, that are proven using other theorems (corollaries) and/or axioms or definitions, such as Fermat's Last Theorem, the ...
5
votes
2answers
288 views

Dogmas and Mathematics

What are the dogmas that restrict or promote the development of mathematics? I know that a dogma is a set of beliefs that is accepted by the members of a group without being questioned or doubted. ...
3
votes
0answers
39 views

looking for good book on the history of formalism

In 1868 Beltrami published a paper ""Saggio di interpretazione della geometria non-euclidea" that seems to have led to the formalist philosophy of mathematics. But what was written exactly what were ...
2
votes
1answer
198 views

Is there a geometrical proof of the impossibility of squaring the circle?

The impossibility of certain constructions in Euclidean geometry, such as squaring the circle with straight-edge and compass is usually shown by using algebraic methods. I am wondering if there are ...
4
votes
0answers
212 views

How much are mathematics driven by applications?

At some point this provocative question came to my mind: Are mathematics mostly driven by applications? I am taking into account some of the comments made to my original question so I want to ...
25
votes
4answers
1k views

Is mathematical history written by the victors?

The question is the title of a recent piece in the Notices of the American Mathematical Society, by twelve authors (of which I am one). The contention is that traditional history of mathematics is ...
3
votes
2answers
247 views

Why demonstrations are important in mathematics? [closed]

Good evening, I'm studying math and would like to know how important are mathematical proofs in the world and particularly in a school of mathematics Thanks for your help
40
votes
3answers
967 views
4
votes
2answers
191 views

Gray's “Plato's Ghost” - a curious mistake

I am currently reading Jeremy Gray's "Plato's Ghost", and I run into the following passage (Chapter 5, page 332). The point is, it seems to me that it contains two very elementary mistakes that feel ...
11
votes
3answers
353 views

What have been some of the most revolutionary philosophical shifts in perspective in mathematics?

Often times, great revolutions in mathematics come from shifts in philosophical perspective. The shift from extrinsic to intrinsic geometry yields manifolds (and much else). The shift in focus from ...
2
votes
1answer
186 views

Did large cardinals exist before 1963?

I'm curious to know the history of the interaction between large cardinals and traveling to (creating) universes through forcing. The question arose because I understand that Peano Arithmatists ...
4
votes
2answers
217 views

why does soundness seem to be less important than consistency for the structuralist?

If I am not wrong, many mathematicians (I believe this is not only restricted to structuralists) agree that an inconsistent formal system does not have any model. By model I mean some kind of set ...
5
votes
10answers
814 views

what is the definition of Mathematics ?

we all study mathematics , and all of us learn mathematical methods to solve problems , we learn how to prove , how to think mathematically but the question is, what is mathematics ? how can we ...
-1
votes
4answers
258 views

The facts about $\varphi$ [closed]

A lot of people believe there is something special about the number $\varphi= \frac {1+ \sqrt5}{2}$. However, I can only think of cultural explanations for looking at each property of $\varphi$ as ...
3
votes
4answers
278 views

Why $\sqrt{\frac {\sum(x-\mu)^2} {N}}$ instead of $\frac {\sum{\Bigl|x-\mu\Bigr|}} {N}$? [duplicate]

Possible Duplicate: Motivation behind standard deviation? In statistics very often you see something of the sort: $$ \textrm{quantity}=\sqrt{\frac {\sum(x-\mu)^2} {N}} $$ to measure things ...
13
votes
4answers
675 views

How do mathematicians think about the existence of numbers?

Question: How do mathematicians think about the existence of numbers? And how did Newton, Euler, and other famous mathematicians thought about this concept? I know that existence of numbers is a ...
4
votes
2answers
262 views

Did Structuralism influence the formulation of Category Theory?

Having only the a very cursory knowledge of Structuralism ( it's a movement generally held to have originated in linguistics, then moving on to philosophy & literature), there does appear to be ...
16
votes
1answer
962 views

What did Gauss think about infinity?

I have someone who is begging for a conversation with me about infinity. He thinks that Cantor got it wrong, and suggested to me that Gauss did not really believe in infinity, and would not have ...
3
votes
0answers
371 views

Mac Lane and Eilenberg's motivations for category theory

I'm looking to understand the conceptual process that brought Eilenberg and Mac Lane in developing the basic concepts of category theory. I quote Mac Lane's book "Category theory for working ...
12
votes
3answers
359 views

Is there any difference between a math invention and a math discovery? [closed]

From wikipekia: The calculus controversy was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented in part by their disciples and associates – ...
14
votes
8answers
1k views

Reference request: is mathematics discovered or created?

I have to write a short monograph as an assignment for a course on the philosophy of science. Being a math student, of course I want to opt for something math-related. After some initial ideas which ...
23
votes
9answers
4k views

Good books on Philosophy of Mathematics

Where can I learn more about the implications, meta discussions, history and the foundations of mathematics? Is Russell's Introduction to Mathematical Philosophy a good start?
0
votes
1answer
155 views

Alternative, consistent frameworks of mathematics with isomorphic mappings to physical phenomenon

A friend of mine who is quite an aggressive Nominalist told me the other day: "Mathematics and numbers are arbitrary; they can accurately predict physical systems in real life only because they are ...