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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
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2answers
261 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 ...
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0answers
86 views

A Question Regarding Ordinal Turing Machines

Consider the following theorem of Koepke: 'A set x of ordinals is ordinal computable from a finite set of ordinal parameters if and only if it is an element of the constructible universe L". Taking ...
4
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3answers
419 views

Visualizing mathematics and geometry

Im writing a paper on the role of visualization in mathematics and specifically geometry. I was wondering if it is possible to represent any arbitrary system of relations and manipulable objects ...
9
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2answers
435 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 ...
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3answers
233 views

Implications and Ordinary language

I studied propositional logic, and everyday I see applications of what I learned on the internet, in mathematical books and miscelaneous resources. One particular case is sentences in the form ...
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4answers
235 views

Is mathematics a science? [duplicate]

Is mathematics a science? I have long considered this to be open to debate, but my professor said that he once heard the quote, ...
9
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2answers
864 views

Who stole the axioms in Natural Deduction?

The study of Gentzen's sequent calculus give me the opportunity to propose some reflections about the concept of logical truth. I'll refer to the english edition of Gentzen's works : The collected ...
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2answers
109 views

About the concept of logical truth

From Frege and Russell to modern mathematical logic textbooks, there were a "shift" of focus from the concept of logical truth, through that of valid formula, to the current concepts of logical ...
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0answers
316 views

Gödel's Completeness Theorem and logical consequence [closed]

At the end of a long process of "rumination" on "old" math log textbooks, I've found the "missing link" - from my personal point of view - between some issues I've raised in the previous months : (i) ...
5
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2answers
121 views

What is gained by internalizing LST (the language of set theory)?

I'm reading up on Gödels constructible universe L in the book "Constructibility" by Devlin, and by comparing his text with texts like Kunen and Jech, there is one thing in particular that he's doing ...
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1answer
149 views

In an infinity of choices, is it possible to guess the correct one?

So I've been thinking about the infinite universes model, where each possible action or event creates a new universe for each outcome. For example, if you flip a coin there will be one universe in ...
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1answer
264 views

Large Operators?

Large operators have always seems strange to me, sometimes their meaning is based on the symbol and other times it has no correlation. For instance, the summation (sigma) has no relation to the ...
0
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1answer
70 views

What is the “lowest” set of axioms that can be used in proofs?

What is the most basic set of axioms that one can use in proofs? As in, the axioms are irreducible. The most basic set of irrefutable rules in mathematics. I assume it has something to do with number ...
0
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1answer
42 views

Reformulation of Theories

Philosophical questions (or even just a matter of taste) regarding some mathematical constructions can give rise to reformulations of whole theories, for example, we can develop (Non-standard) ...
1
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2answers
350 views

What problems are there left to solve? [closed]

From the ancient Greek mathematicians (Archimedes, Pythagoras) before Christ to Issac Newton to George Birkhoff, these mathematicians have made huge strides in mathematics, developing theorems and ...
5
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3answers
141 views

On trusting the mathematical process [closed]

In studying math we are, at least partially, interested in making abstraction of real world problems and solving them through rigorous techniques and methods, and then interpreting the result. Let us ...
12
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2answers
265 views

Founding Arithmetic on geometry

In the past I found some fleeting references that some (Frege in his later years being one of them) tried to found arithmetic not on set-theory and logic but on geometry and logic. Unfortunedly Frege ...
3
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2answers
247 views

Again about McGee objections to modus ponens

I would like to "reopen" the previous post regarding Modus ponens because, frankly speaking, I'm not satisfied with some (most of ?) answers by the mathematicians community. Disclaim: I'm not aiming ...
0
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1answer
94 views

Finality of mathematics [duplicate]

A random question came to me, which looks something like this : Is there such a thing as a "finality" of mathematics ? What I mean is can we imagine a time where there would be no more mathematics to ...
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2answers
103 views

How is it possible for something to be less then nothing? [duplicate]

What is the ontological state of negative numbers? Is it a human invention or a does it live with reality?
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6answers
300 views

What is the right interpretation of the axiom of extensionality

A set $a$ can be called extensional if it has the following propery: $$\forall b\left[\forall x\left[x\in b\iff x\in a\right]\Rightarrow a=b\right]$$ Based on this the axiom of extensionality can be ...
3
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3answers
331 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). ...
2
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1answer
123 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 ...
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3answers
68 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 ...
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4answers
2k views

how do we assume there is infinity?

Definition of infinite: A set is infinite iff it is equivalent to one of its proper subsets. We know that our universe doesn't contain infinite number of elements (including subatomic particles), so ...
7
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1answer
390 views

Why can almost all ordinary mathematics be formalized by sets?

there must exists a reason of why the idea 'collection' is so powerful that it can formalize nearly all mathematics. subquestion: is there any which can not be formalized by this perspective? if so, ...
3
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2answers
191 views

Non-well-founded models viewing well-founded models as non-well-founded.

I'm currently thinking about how different models of set theory view each other. In particular I'm looking at how well-foundedness behaves between different models. So we have the Axiom of ...
4
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2answers
168 views

Is Paraconsistent Negation Really Negation?

Let a logic be paraconsistent, if $\phi \wedge \neg \phi \not \models \psi$ for some $\phi, \psi$ (where $\models$ is the logic's consequence relation). There are different ways to prevent a ...
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3answers
720 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 ...
2
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1answer
161 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 ...
3
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2answers
123 views

How do we distinguish between characteristic 0 and characteristic p for very large p?

This is a somewhat soft question, apologies if it turns out to be trivial/nonsensical. Background: I was half-asleep one morning, not quite through my first cup of coffee, and thought about the ...
6
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4answers
641 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] ...
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3answers
2k 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. ...
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3answers
385 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
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3answers
246 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 ...
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2answers
161 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 ...
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1answer
51 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 ...
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2answers
287 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 ...
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4answers
279 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 ...
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2answers
163 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 ...
5
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1answer
120 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
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1answer
89 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 ...
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1answer
192 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.
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2answers
341 views

How come mathematics is applicable to the real world?

Often in mathematics one constructs a set of some sort, let's name it $A$. We've constructed it in an abstract way, so, a priori, structural aspects of $A$ are yet unknown to us, until we prove them. ...
4
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2answers
216 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.
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1answer
188 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' ...
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1answer
274 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 ...
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76 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 ...
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1answer
156 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 ...