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1answer
81 views

Can we take definability and existence as primitive notions of a theory?

One of my friend tries to develop an alternative viewpoint of Set Theory. For this he has taken the terms binary relation, set, existence and definability as primitive notions of his Set Theory. After ...
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4answers
167 views

How do we know logic works? [duplicate]

Every time I read about a theory in mathematics, it usually starts with axiomatizing the most fundamental concepts that are going to be treated. Recently, I have started finding this troubling. In ...
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2answers
81 views

Are theorems like subroutines for math? [on hold]

I've been developing more appetite for math just lately, as I study electromagnetics to deepen my understanding of electric circuits and devices. I'm finding that doing derivations as exercises helps ...
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0answers
38 views

Is it possible to be a frequentist and a subjectivist at the same time?

I'm trying to understand the differences between (1) Bayesian vs frequentist; and (2) subjectivist vs objectivist. So far my understanding (correct me if I'm wrong) is that: (1) Bayesian vs ...
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1answer
969 views

What Do Mathematicians Do?

The American Mathematical Society maintains a web page entitled "What Do Mathematicians Do?" which references two interesting surveys. (One of the reference links is broken, but this one works: What ...
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3answers
77 views

Basic atoms in mathematics [closed]

Given the concepts '1', 'set' and 'sum' (and maybe 'point' for geometry), can you build the whole mathematics upon then? If not, what other basic atoms would you need?
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3answers
79 views

Infinite sums vs infinite unions

Why is it that For every set $S$, there exists a set $\bigcup S$. is something we take for granted (even though $S$ could be infinite), while For every sequence $a_1,a_2,\dots$ of numbers, ...
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9answers
1k views

Is math an exact science? [closed]

The day before yesterday I talked with a friend of mine about math. He is also a PhD student like me, and in his opinion math cannot be consider an exact science, because the same statement could be ...
3
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1answer
39 views

Differences between constructivism and formalism

What are the main differences between the formalism and constructivism in mathematics? Is there some theorem or axiom valid in formalism which isn't valid in constructivism and vice versa? Is the ...
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2answers
173 views

How different are the positive and negative numbers?

Is there a fundamental difference between the positive and the negative numbers? Or is the difference like the one with electric charges in physics, where the other type of charge was just decided to ...
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2answers
71 views

In “10 grams of salt”, is the unit “grams”?

The gram is a unit of mass, so "10 grams" has "grams" as the unit. "10 pounds" uses a different unit. So what is the "salt" in "10 grams of salt", if not a unit? In other words, what is the ...
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0answers
59 views

Any “exotic” foundations of mathematics? [closed]

There is a myriad of axiomatizations of set theory (a branch of mathematics obviously not at all identical with notorious ZFC) and other formal systems working with classes, categories and such. All ...
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3answers
55 views

Why does $p$ (is true) strictly agree with $p$ while $p$ (is false) strictly disagrees?

Let's make the truth table: $$\begin{array}{|c|c|c|} \hline p&(p) \text{ is true}&(p) \text{ is false}\\ \hline T&T&F\\ F&F&T\\\hline \end{array}$$ "$p$ is true" strictly ...
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8answers
4k views

What does it mean to solve an equation?

This question might be more philosophical than mathematical. In school we are taught how to solve equations such as $x^2 - 1 = 0$ or $\sin(x) - 1= 0$. Solutions to these equations are quite simple. ...
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2answers
56 views

Logical explanation why exponentiation operation is not commutative and associative

Considering Peano axioms we'll define addition, multiplication and exponentiation operations. We can then prove that addition and multiplication operations are commutative and associative. The proof ...
3
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1answer
49 views

If a theory $A$ can prove $B$ consistent relative to $A$, and $A$ is consistent, does $B$ have to be consistent?

Let's say we have two sets of axioms $A$ and $B$ such that $\mathsf{ZF} \subseteq A \subseteq B$, and from $A$ we can prove that if $A$ is consistent, then $B$ is consistent as well (that is, $A \...
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2answers
68 views

Find the divisors of $5040$ in the Plato's dialogue “Theaetetus”

In the Plato's dialogue "Theaetetus", at a certain point, we have the following "problem" \begin{align*} 5040 &= 7! \\ &= 1\times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \\ &= 2 \...
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11answers
1k views

Problems that are largely believed to be true, but are unresolved

Are there unsolved problems in math that are large believed to be true, but for reasons other then statistical justification? It seems that Goldbach should be true, but this is based on heuristic ...
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10answers
5k views

Is it an abuse of language to say “*the* integers,” “*the* rational numbers,” or “*the* real numbers,” etc.?

I'm finding that the more math I learn, the more concepts I thought were well-defined seem to be intuitive and naive. Here I'm asking about whether it's an abuse of language to refer to "the integers,"...
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0answers
26 views

Multiplying an object by time instead of dividing. What happens when?

Let "a" be apples, and "s" be seconds. a= 1 apple. a/s= 1 apple for every second (an accumulation of apples). a/s/s= 1 apple for every second, for every second (an acceleration of the accumulation ...
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2answers
1k views

What's going on with this 5-line proof of Fermat's Last Theorem? [duplicate]

I'm reading a book on the Philosophy of Mathematics, and the author gave a "5-line proof" of Fermat's Last Theorem as a way to introduce the topic of inconsistency in set theory and logic. The author ...
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4answers
1k views

What would happen if ZFC were found to be inconsistent?

If, one fine day, someone found a contradiction in ZFC (or even ZF), what implications would such an event have for mathematicians? Is there currently any backup axiomatic system on par with ZFC that ...
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0answers
55 views

How to become fluent at reading math formulas? [closed]

As part of my studies, oftentimes I need to read research publications which contain mathematical formulas. Whenever I have to do that, I feel discouraged. Somehow I can not comprehend the ...
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2answers
129 views

When are quantities outside of the real numbers considered equal, and when do they exist?

I know of the complex number $i$ and it's existence as the result of invalid square rooting (the square root of negative one does not exist inside the real numbers), but other than complex numbers, ...
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1answer
50 views

The positive introspection axiom

I am studying modal logic with the textbook 'Reasoning about Knowledge' Fagin et al. 1995 The positive introspection axiom is taken as something that can be proved with the possible worlds model of ...
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2answers
50 views

Necessary truth of mathematical proposition.

Take from Possible world- an introduction to logic and its philosophy. p-21 Following quote provide us with necessary definition of what "logically necessary" or as far as i think "necessary truth" ...
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6answers
360 views

Do different notations imply different properties of a number?

I had an argument with a friend of mine and I'd be glad if someone could clarify things a little bit. So, let's say we have an integer, eight or seventeen, for example, doesn't matter. It has all the ...
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1answer
72 views

A philosophical question on probability theory [closed]

This question is philosophical in nature. The example is taken from theology, but one may invent more examples, including these more scientific than mine. Nevertheless it is a valid mathematical issue....
2
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1answer
70 views

“Concrete” Realisations of non-abelian finite groups

Many commutative groups could be imagined as something "concrete", for example $\mathbb N$ as an abstraction of operations on sets of objects, and from this $\mathbb Z, \mathbb Q$ and $\mathbb R$ are ...
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4answers
699 views

What is… A Parsimonious History?

Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a ...
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0answers
25 views

Tensors as mathematical objects

Continuing my journey to understand Tensors, Maxwell's equations. Here is my current understanding. Is it correct? Tensors are mathematical objects, i.e., an entity in mathematical reality or a ...
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6answers
1k views

Meaning of the word “axiom”

One usually describes an axiom to be a proposition regarded as self-evidently true without proof. Thus, axioms are propositions we assume to be true and we use them in an axiomatic theory as premises ...
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1answer
96 views

Is it circular to define the Von Neumann universe using “sets”?

I was just reading the Wikipedia page on the Von Neumann universe, where it is stated that this universe "is often used to provide an interpretation or motivation of the axioms of ZFC." However, later ...
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1answer
289 views

When is a function a dimension?

The concept of dimension is used in many different contexts. Generally a dimension is a function that has as domain some family of sets ad has value on a set that, in the most common situations, is $...
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1answer
62 views

Cauchy's real line and math philosophy till XIX

I have to write an essay concerning philosophy of mathematics until the end of $XIX$ century. I've heard that the reason why the Cauchy's theorem (if continuous functions $f_n \rightarrow f$ then $f$ ...
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3answers
146 views

On “why” questions in mathematics

In response to the question How would one be able to prove mathematically that $1+1 = 2$?, Asaf Karagila explains: In a more general setting, one needs to remember that $0,1,2,3,…$ are just ...
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0answers
69 views

Is Leibnizian calculus embeddable in first order logic?

We just published an article making what we feel is a plausible case in favor of an affirmative answer in Foundations of Science, see preprint here. The basic argument is that while such a requirement ...
2
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1answer
51 views

Have humans ever used the Log Scale convention in the past rather than the Linear one?

There are many examples where our senses are based off of log scales such as volume of a noise, ability to guess (i.e.) plus or minus a power of 10 with Fermi, and even when we measure pain on 1 to 10 ...
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2answers
83 views

On logic vs information theory

If the statements All crows are black and All non black things are non crows are equal, then why is the former so much easier to communicate by giving examples? What implications does this ...
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9answers
5k views

Is complex analysis more “real” than real analysis?

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
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2answers
170 views

Law of Excluded Middle Controversy

I was reading an introductory book on logic and it mentioned in passing that the Law of Excluded Middle is somewhat controversial. I looked into this and what I got was the intuistionists did not ...
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0answers
55 views

Can I just make this function up?

The Lambert W function was made to solve the problem $xe^x=k$ for $x$, which is given as $x=W(k)$. Could I just make a function $x=F(k)$ which solves $x\cos(x)=k$? Even though the solution has an ...
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9answers
2k views

Is formal truth in mathematical logic a generalization of everyday, intuitive truth?

I'm trying to wrap my head around the relationship between truth in formal logic, as the value a formal expression can take on, as opposed to commonplace notions of truth. Personal background: When I ...
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9answers
6k views

Does mathematics require axioms?

I just read this whole article: http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf which is also discussed over here: Infinite sets don't exist!? However, the paragraph which I found most ...
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4answers
491 views

The standard role of intuitive numbers in the foundations of mathematics

In my career I've been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory. However, I am not quite sure about ...
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0answers
71 views

Are all statements about math inherently formal? Can one do math without formal logic? [duplicate]

Are all people who do mathematics applying (whether they know it or not) formal logic? Does every statement someone may make about math, at its core, a formal statement in mathematical logic? (I'm ...
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1answer
46 views

Different Mathematics

Hey I am a high school student who is very interested in the philosophy of mathematics. I was watching this talk by Stephen Wolfram about whether or not mathematics is invented or discovered. In it he ...
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35answers
43k views

Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
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4answers
105 views

Should we or should we not take $1$ as a prime number? [duplicate]

I think I know that there were times in the past when it was convenient to look at a number $1$ as a prime number, and, as far as I can remember, even then it was dependent on who we ask is it prime ...
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1answer
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A question about the real line and the Dirichlet function.

Though the graph of the Dirichlet function is non-drawable, I think if we have to draw it in some informal way then it will be two complete lines (instead of isolated points). Here's my reasoning: ...