Questions involving philosophy of mathematics

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15
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3answers
370 views

Mathematical Notation and its importance

You can see how mathematical notation evolved during the last centuries here. I think everyone here knows that a bad notation can change an otherwise elementar problem into a difficult problem. Just ...
1
vote
1answer
134 views

What concepts does math take for granted?

I suspect there must be some concepts that math takes for granted (there has to be a starting point). For example, after spending some time thinking about it yesterday, I wondered whether most of ...
5
votes
4answers
432 views

What is the “correct” reading of $\bot$?

I have some doubts about the "natural" interpretation of $\bot$ in Natural Deduction and sequent calculus. In Prawitz (1965) $\bot$ (falsehood or absurdity) is called a sentential constant [page 14] ...
7
votes
3answers
684 views

Are all mathematical statements true or false?

I would like to know whether it can be possible for a statement to be neither true nor false. Consider the age old paradox. "This statement is not true" Clearly it cannot be true. If it is false. ...
4
votes
3answers
340 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 ...
3
votes
3answers
169 views

Provocations on the existence of mathematical objects

The few Mathematics I have been studying so far is pure Mathematics. I happen to have some discussions with philosophers of Mathematics, but as they know I totally ignore their subject, we do not ...
1
vote
2answers
148 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 ...
1
vote
1answer
43 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 ...
10
votes
2answers
213 views

When the mathematical community consider the inclusion of a new axiom?.

At first I was thinking about the axiom of choice, but let's keep it general. What motivates the inclusion of new axioms (or change the ones we already have in an already defined axiomatic theory?. It ...
4
votes
4answers
236 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 ...
0
votes
1answer
116 views

Does Graphical evicence count as / contribute to a Proof in Mathematics?

Several questions such as the following have an answer with pictures in it. How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$ How prove this inequality ...
4
votes
1answer
88 views

What is the intuition behind $\Delta_1^0$ sets and $\Delta_1^1$ sets?

In the context of first-order arithmetic, if $\phi$ is a formula with only bounded quantifiers, then if you put existential quantifiers in front it becomes a $\Sigma_1^0$ formula according to the ...
2
votes
1answer
84 views

Philosophical side of MATH. knowing the path then walk it. [closed]

Can I find a book that gives me the purpose of theorems and definitions without going deep into proofs. It's just like knowing the path then walk it. That's will me the understanding reach the next ...
6
votes
1answer
178 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.
3
votes
3answers
287 views

How come mathematics is applicable to the real world?

Before the dam breaks, namely, the one that holds the waters of accusations, I want to specify that the question I'm asking is a "reference-request", and therefore does have an answer. Often in ...
4
votes
2answers
203 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.
2
votes
1answer
169 views

What's with conditionals in mathematical logic?

Having a bit of difficulty understanding the conditional ($\rightarrow$) in mathematical logic. I read up on the already-existing questions and it did help me understand it better (the 'promise' ...
5
votes
1answer
175 views

What is the difference between ❋3.01 and ❋4.5 in Whitehead and Russell's PM?

This baby step from ❋3.01 to ❋4.5 is so tiny that I can barely see the difference. Please kindly explain why it is so important to distinguish the two. What is the philosophical importance of this ...
3
votes
0answers
73 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
118 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 ...
2
votes
5answers
381 views

What is maths? “Maths is the study of ______”? [closed]

I can fill in the blank by just listing the different fields of maths but my goal is to define all of mathematics. An answer that I would've accepted a few years ago is "Maths is the study of ...
2
votes
1answer
114 views

Is it possible for us to know something to be true without actually proving it? [closed]

I know, proof is the most crucial part of mathematics, it makes all the things be rigorous and keeps mathematics from contradiction. In real life, there's things that we know to be true, for sure. ...
8
votes
1answer
148 views

Do second-order categoricity proofs require a background concept of set?

In his article "The Set-Theoretic Multiverse", Joel David Hamkins (as part of his reply to Donald Martin's argument that the set-theoretic universe is unique, found in "Multiple Universes of Sets and ...
1
vote
0answers
68 views

Are there intensional classes independent of the set universe?

The hereditarily finite sets can be regarded as purely extensional sets. Furthermore, they are quite independent of the underlying set universe (at least if we look at them from an extensional point ...
5
votes
1answer
81 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 ...
-1
votes
1answer
144 views

Why is the number Pi more popular than any other constant? [closed]

What is so special about the number $\pi$? There are many more interesting constants, such as e, $\gamma, \sqrt{2}$ or the catalanian number. $\pi$ has been calculated to more digits than any other ...
1
vote
1answer
501 views

Abductive v.s. inductive reasoning

To me, abductive reasoning and inductive reasoning are very very similar, in that they both go from the specific to the general and they are distinguished only through the examples which are provided ...
3
votes
1answer
119 views

Can equinumerosity by defined in monadic second-order logic?

Two properties (or concepts) $F$ and $G$ are said to be equinumerous if they have the same cardinality, i.e. if they can be put in one-to-one correspondence with each other. This can be very easily ...
12
votes
1answer
386 views

What underlies formal logic (or math, generally)?

I read a useful definition of the word understanding. I can't recall it verbatim, but the notion was that understanding is 'data compression': understanding happens when we learn one thing that ...
3
votes
2answers
284 views

Leibniz' Law and that good old riddle

There exists a Theory of Identity in mathematical logic. I've encountered it for the first time in Principia Mathematica by Alfred North Whitehead and Bertrand Russell (1910). Quote: "This definition ...
0
votes
2answers
120 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 ...
-4
votes
2answers
322 views

Do irrational numbers really exist?

Isn't it possible that an irrational number is in reality the quotient of two infinitely long integers that even if there were repeating sections in it, it would take infinite digits to find the first ...
3
votes
2answers
193 views

Understanding: Axiom of Specification and Russell's Paradox: there is no universe?

Following Halmos's Naive Set Theory, Russell's Paradox emerges from using the axiom of specification (that for every set $A$ and property $\phi$ there exists a set $Y$ whose elements are those ...
5
votes
1answer
181 views

Explain/illustrate Goedel's theorems and possible implications to non-mathematicians

I am asked to give a talk about (a) mathematical practice, (b) axiomatization, (c) Gödel's theorems and (d) possible antimechanist arguments based on the incompleteness theorems (as mentioned in P ...
1
vote
1answer
161 views

Explain mathematical practice and axiomatization to non-mathematicians

I am asked to give a talk about (a) mathematical practice, (b) axiomatization, (c) Gödel's theorems and (d) possible antimechanist arguments based on the incompleteness theorems (as mentioned in P ...
4
votes
2answers
213 views

Why do imaginary numbers work (somewhat philosophical question)?

Asking as a layman, I've always puzzled over imaginary numbers and how they can be used to solve problems involving real numbers or quantities only (e.g. contour integration methods or Fourier ...
11
votes
1answer
390 views

What would qualify as a valid reason to believe there is a closed form?

I noticed that almost every non-homework-level integral posted on this site prompts somebody to ask "Do you have any reason to believe there is a closed form?" (some recent examples here and here) I ...
5
votes
1answer
747 views

How much math does one need to know to do philosophy of math?

I'm looking for advice from mathematicians who also study philosophy of math (PoM). Due to interest I'd like to study PoM as a hobby, but I'm worried if I don't understand math well enough from a pure ...
2
votes
2answers
195 views

Is there a proper term and/or symbol for an “agnostic” conclusion?

My question stems from the material conditional: $p \rightarrow q\\p\\\therefore\space q$ However, if $\bar p$ then the conditional is silent. I would like a way to represent this fact using, if ...
0
votes
2answers
141 views

Does infinity have a limit?? [closed]

Infinity being the far extent that the numerical system can stretch,can we say that infinity is actually a limit or infiity has another limit?
70
votes
22answers
6k views

Is math built on assumptions?

I just came across this statement when I was lecturing a student on math and strictly speaking I used: Assuming that the value of $x$ equals <something>, ... One of my students just rose ...
6
votes
5answers
326 views

Are there more real numbers than we can actually imagine?

I mean, if we could imagine all the real numbers then we could assign each number a finite sentence (or a finite book). Since the set of the finite books is countable then the set of real numbers ...
5
votes
2answers
318 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. ...
0
votes
5answers
337 views

Why doesn't $1/x=0$ have any solution?

Just out for curiosity ! Why $1/x=0$ doesn't have any solution? Or is it that the solution takes you to $1=0$ situation which would nullify mathematical principle that we stood for years Educate ...
92
votes
17answers
12k views

Is 10 closer to infinity than 1?

This may be considered a philosophy but is the number "10" closer to infinity than the number "1"?
3
votes
0answers
42 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 ...
1
vote
1answer
106 views

Analogue of prime numbers in addition? [closed]

What is the analogue of prime numbers in addition?
2
votes
1answer
194 views

True and provably true sentences in a model. Are they the same thing?

In logic, it is said that each sentence in a (consistent) theory is either true or false in a given model. Checking the truth of a sentence in a finite model amounts essentially to finite enumeration ...
3
votes
0answers
115 views

Can one define informational content of a mathematical expression?

At least in physicist's thinking, information, vaguely, is something that allows one to select a subset from a set. Say, a system can be in states A and B, we have done a measurement on it ...
1
vote
3answers
113 views

how can we write abstract algorithms?

Writing pseudo-code for algorithms is common practice in the applied mathematics literature. It is also often the case that the ideal input of an algorithm is an infinite set, for example it could be ...