Questions tagged [philosophy]

Questions involving philosophy of mathematics. Please consider if Philosophy Stack Exchange is a better site to post your question.

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What is “ultrafinitism” and why do people believe it?

I know there's something called "ultrafinitism" which is a very radical form of constructivism that I've heard said means people don't believe that really large integers actually exist. Could someone ...
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1answer
132 views

Is there criticism in literature of Euclid's fifth common notion (“The whole is greater than the part”)?

In Book I of Euclid's Elements, the fifth common notion says "The whole is greater than the part". For Euclid, magnitudes are objects that can be compared, added, and subtracted, provided they are of ...
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1answer
31 views

What is the minimum number of things needed to declare you have a variety? [closed]

When people say things like "we have a wide variety of products" or "product x can run in a variety of modes", what is the lowest number of modes or products which one can comfortably call a variety? ...
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1answer
69 views

Why is it possible to use logic? [closed]

First of all: there is a philosophy of mathematics tag in this SE, so this could be the proper site for my question. I do not know how to evaluate this. When we assume an axiom to prove something, ...
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What is a fair coin?

The title of this question is almost a retorical question. My point is that there is no way to define probability in a non circular manner. Let's say the probality of getting a tail when tossing a ...
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Arithmetic systems without Induction

It's often said that AC is a controversial axiom and so often in my math classes when it is used a brief comment is made to the effect of "we can prove this without Zorn's Lemma but it's more work". ...
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41 views

Proof in mathematics vs everyday [closed]

Does the word "proof" has meaning only in mathematics ? when in conversations someone ask someone else to prove something what exactly does he mean ?
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2answers
45 views

Sorites paradox natural deduction problem

I'm new to natural deduction and am attempting a question which sounds like the Sorites paradox: 1 million grains of sand is a heap. If 1 million grains of sand is a heap, then 999,999 grains is a ...
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1answer
86 views

What are the arguments of the mathematicians who objected against the ontological proof Gödel offered?

Q: What are the arguments of the mathematicians who objected against the ontological argument/proof Gödel offered? $$ \begin{array}{rl} \text{Ax. 1.} & \left\{P(\varphi) \wedge \Box \; \forall ...
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1answer
101 views

Hilbert Program: Consistency vs Soundness

I have been wondering why exactly Hilbert asked for a decidable, complete, and consistent set of axioms for mathematics, rather than a decidable, complete, and sound set of axioms. Consistency being: ...
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Why do we need to learn Set Theory?

I was planning to write some article for the Mathematics magazine of our college and it occurred to me that it will be a good idea to write about the impact and importance of Set Theory. I plan ...
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5answers
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Can mathematics get from other sciences what it got from physics?

Throughout history, physics has been an unparalleled source of '' inspiration'' for discovering/inventing mathematical ideas, which is due to its ability to describe the physical world. But can this ...
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1answer
112 views

What formal proof systems are capable of proving $\forall x \exists y x = y$ without needing to apply $\forall$-I to $\exists y x = y$?

I am interested in some philosophical questions that depend on whether the open formula $\exists y x = y$ is a logical truth. I'm making the assumption that some logical systems are intended, in the ...
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2answers
237 views

Books on the philosophy of geometry

I am looking for recent books ( say published after 2000) on the philosophy of geometry, most books on the philosophy of mathematics seem to ignore or bypass geometry at all or am I just looking with ...
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2answers
136 views

Is there any “good” definition for what constitutes “applied mathematics”?

Is there any "good" definition for what constitutes "applied mathematics"? Wikipedia lists stuff such as statistics, optimization. However, e.g these have certainly "pure mathematical" aspects to ...
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44 views

What is the relationship between the common concept of “model” and “model” as used in Model Theory? [duplicate]

To my understanding, a model in Model Theory is an interpretation (in a form of a set or other algebraic structures) for a certain sentence S which makes S true. In everyday language, and also in ...
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1answer
45 views

Is there an accurate view on the distinction between “what mathematics can model” and what it cannot? [closed]

Is there an accurate view on the distinction between "what mathematics can model" and what it cannot? Not just in hard sciences. What about social, political questions? What's the accuracy of ...
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8answers
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How irrational quantities physically exist in nature?

We know that an irrational no has well defined decimal values upto infinite decimal places. These irrational quantities exist in nature in some kind of measurements. For an example, circumference of a ...
4
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3answers
1k views

Is “This sentence is true” true or false (or both); is it a proposition?

From what I understand, a proposition is either true or false, but not both. "This sentence is false" can be neither true nor false and is thus not a proposition. However, is "This sentence is true" ...
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0answers
61 views

Why we need such a restrictions in logics?

Note:I am not competent is logic so this question may look weird to you. So as I know there are different types of logics (first-order logic, second-order...), and the difference between them is that ...
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1answer
155 views

Why do we need/use proof theory?

Note that my knowledge of both proof theory and model theory is incredibly weak. I just started learning about them using Kleene's "Mathematical logic". If I understand it correctly then one of the ...
2
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1answer
158 views

Is mathematics a syntax?

I have read that syntax is symbol and semantics is meaning those symbols convey. Is mathematics a syntax? Where is semantics in mathematics? What gives meaning to mathematics, to those symbols I ...
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1answer
412 views

Choosing formal system for mathematics

I always wondered, we have many choices for choosing what kind of postulates - axioms, deduction rules, we choose for our formal system. For example, there are Hilbert style systems where there are ...
3
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1answer
131 views

Understanding impredicative definitions [closed]

In studying more on the mathematics in the past of Frege, Russell, and Zermelo, and I was wanting to learn more about impredicative/predicative definitions to solve some inquiries I had. 1. How does ...
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1answer
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Understanding a quote from G. H. Hardy in 'A Mathematician's Apology'

I recently learned about the philosophy of constructive mathematics. In several discussions of the topic, I keep seeing a quote from G. H. Hardy's book A Mathematician's Apology; Reductio ad ...
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3answers
68 views

A friendly introduction to Hilbert's sixth problem with special focus in its meaning and issues

While I was reading the Wikipedia's article dedicated to Hilbert's problems I've known the so-called Hilbert's sixth problem from this page and the corresponding Wikipedia's entry dedicated to this ...
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3answers
275 views

Specific axioms under which the continuum hypothesis is true or false

It is well-known that the continuum hypothesis (CH) cannot be proven under the standard axioms (i.e. independence from ZFC). However, to (non-expert, beginning student of the field) me, it seems ...
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2answers
215 views

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

From the context of the real numbers we know that the square root of $-1$ is not a real number. Because of this the historical way we approach this is that the concept of complex numbers were created ...
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1answer
59 views

Negation-incompleteness, Godel's theorems and interpretations

So, as far as I understand, Godel's theorem says that sufficiently strong theories such as Peano Arithmetic are negation-incomplete, which means that there exists a formula $\phi$ such that neither $\...
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0answers
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I'm having difficulty understanding this problem to do with statistical inference (specifically, Bayesian) in scientific investigation.

I'm over here from the philosophy page since a very similar question that I asked there a couple times wasn't ever properly answered and I think statisticians here might be able to provide a helpful ...
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1answer
373 views

Proving a Mathematical hypothesis using Physics

We know that mathematical models of physical phenomena need to becomes more and more sophisticated as our observations become more precise and comprehensive by utilizing more advanced instruments that ...
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1answer
66 views

Why we can safely treat objects like mathematical entities?

In the study of number systems we learn the axioms of real numbers. For example: The commutative axiom $X.Y = Y.X$ The distributive axiom $X.Z + Y.Z = (X+Y).Z$ Well, the thing is that we are ...
179
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4answers
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How do I convince someone that $1+1=2$ may not necessarily be true?

Me and my friend were arguing over this "fact" that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical ...
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6answers
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Why accept the axiom of infinity?

According to my readings, Russell showed that a principle Frege used to reduce Peano arithmetic to logic lead to a contradiction. So, Russell tried to reduce mathematics to logic a different way but ...
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5answers
712 views

Peano's Axioms: Mathematical Philosophy

In Peano Axioms, why is it necessary to define number and successor. Does not using them imply that we know what they mean? Or could they have just as easily been any two arbitrary terms which are not ...
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0answers
101 views

Motivation of the Continuum Hypothesis

Why do mathematicians care about the Continuum Hypothesis? Does it have philosophical implications? If it was true or false, would it have had some sort of implications in mathematics? Does the ...
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2answers
127 views

Do all mathematical ideas eventually find their way into the real world?

I am writing an essay for a scholarship and they asked the following question: "How will these goals enable you to help others?" Although I am majoring in pure math, this really got me to think about ...
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1answer
258 views

How do we know we get the right answer?

The problem of ontology is one much discussed in mathematical philosophy with much categorization into different schools of thought, but the problem of epistemology seems to be less discussed; ...
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1answer
65 views

Why did Frege need to use Courses-of-Value in his number Concept

I am studying Frege's work and Russell's Paradox. I can't understand why did Frege need to use Courses-of-Value in his number Concept, wouldn't it be enough to state "Any concept F is equal to Zero, ...
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3answers
2k views

Why Maximize Expected Value?

In many instances I've come across (in Game Theory, etc), when trying to choose an optimal strategy it has the criterion that it wants to maximize expected value much of the time. To simplify this ...
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1answer
105 views

Induction on complexity, upside down?

I have to prove that in a formation sequence of a formula F, all formulas that appear are sub-formulas of F. The proof that the text (Boolos, Computability and Logic) suggests is by induction on ...
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2answers
303 views

Why and how is logic related to set theory?

I am learning set theory on my own at the moment, and I realised I can't avoid not to learn logics. There is a strong connection between these two. such that, proofs for sets are based on logics. I ...
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1answer
250 views

How do set theorists view this issue?

Since the question has changed significantly over the course of last few days, it was suggested to modify it in that light. I have removed those parts which aren't directly related to the main ...
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5answers
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Evidence of Absence = Absence of Evidence?

Any clever-cloggs out there who can explain the formula below in more simple English please? - Do you agree with the formula?
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The understanding of category of groups

When we say the category of groups, do we refer to the same category of groups? For example, in Tom's mind, there is a category of groups; in Jack's mind, there is a category of groups. However, one ...
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1answer
47 views

Are predicates merely convenient or are they necessary in logical language?

Model theory is typically based on a formal language whose production rules and syntax are designed to formalize the deductive process, and as such typically include predicate or relation symbols. ...
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1answer
61 views

Assumptions necessary to justify the method of proof by contradiction.

It seems to me, these assumptions are necessary to demonstrate proof by contradiction: i) Every proposition must belong to $T$ or $F$. ii) No proposition belongs to both $T$ and $F$ iii) If having $...
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8answers
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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 ...
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65 views

Can we be sure proofs have no errors?

My current understanding is that work submitted to journals has mathematicians look over it for errors. Mathematics is deductive, yet with this being the burden of proof, how can we know for sure ...
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1answer
45 views

About proofs that we cannot verify every step by hand

For something I am planning to write, I need to clarify few issues with respect to computer-aided proofs that we cannot verify every step by hand. For example, the proof for the four-color map theorem....