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Number of symbols required to represent a number in unary?

I was thinking about the different number systems, and realised that technically binary is not the simplest. The simplest is unary - i.e. powers of 1. Wikipedia confirms this view: https://en....
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0answers
63 views

How to apply rigor to proof sketches

When doing analysis, I have difficulties to recite the proofs given in the lectures in oral examinations. I am able to do calculations with the theorems that I encountered. (Note that in central ...
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2answers
157 views

Is there any areas of mathematics that depend on the continuum hypothesis being true? (Or not)

I've been looking up a little bit about the Continuum Hypothesis, I understand that it is independent of $ \textbf{ZFC} $, that is, cannot be proved or disproved using the current axioms in $ \textbf{...
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8answers
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What's your explanation of the Raven Paradox?

The Raven Paradox starts with the following statement (1) All ravens are black. which is equivalent to the following statement (2) Everything that is not black is not a raven. In all the ...
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2answers
406 views

Understanding Hilbert's program

Upon covering Hilbert's program in class and consulting https://plato.stanford.edu/entries/hilbert-program/, there were some questions I had on Hilbert's work, and was trying to find answers to them. ...
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4answers
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How are primitive mathematical objects chosen?

On page of 22 Terry Tao's Analysis I, he writes (after presenting the Peano axioms): Remark 2.1.14. Note that our definition of the natural numbers is axiomatic rather than constructive. We have ...
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1answer
49 views

a modal logic question

I am looking for help understanding a problem of modal logic. In his 1963 essay 'Hume on evil', the philosopher Nelson Pike raises the following inconsistent triad of propositions (where S, W, and L ...
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149 views

Intuitive Randomness, Uniform PDFs, and Bertrand's Paradox

Suppose that somebody asks us the following question: Consider a straight line segment that goes from 0 to 1 (inclusive). Suppose that a point is chosen at random on this line segment. What is ...
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3answers
467 views

Natural language examples for failure of double negation elimination

I am trying to explain why double negation elimination $\neg \neg \phi \vdash \phi$ is invalid in intuitionistic logic, but introduction $\phi \vdash \neg \neg \phi$ is valid. The latter is easy: ...
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5answers
843 views

How does one understand and resolve Zeno's paradox?

Zeno, a follower of Parmenides, reasoned that any unit of space or time is infinitely divisible or not. If they be infinitely divisible, then how does an infinite plurality of parts combine into a ...
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1answer
225 views

Is this the basic loophole in Zeno's paradox?

So, Zeno assumes that, to go from the mark at $1m$ to the mark at $2m$ we've to do an infinite number of tasks. Like the task of getting to $1.001m$, the task of getting to $1.000005m$,the task of ...
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1answer
108 views

Linear Temporal Logic - Combining Formulas

So the intent is to construct an LTL formula that describes the following scenario: There are two agents, Anastasia and Boris, who have arranged to meet at a cafe and exchange secret information. ...
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What does “most of mathematics” mean? [closed]

After reading the question Is most of mathematics not dealing with sets? I noticed that most posters of answer or comments seemed to be comfortable with the concept of "most of mathematics". I'm not ...
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3answers
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Is most of mathematics independent of set theory? [closed]

Is most of mathematics independent of set theory? Reading this quote by Noah Schweber: most of the time in the mathematical literature, we're not even dealing with sets! it seems that the answer ...
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4answers
680 views

Why is the distributive property so pervasive in mathematics?

I just read this post which gives a geometric argument for the distributive law for real numbers, which I liked: https://math.stackexchange.com/a/466397/241685 However the distributive law comes up ...
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0answers
35 views

Game theoretic foundation of Kantian imperative like thinking

Let's say I'm involved in some project (e.g. humanity) in which I can invest some time, but I have no direct information about the amount of time spent by the other project members. Some common sense ...
2
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1answer
204 views

Linear Temporal Logic - Distributivity of Temporal Modalities

So in LTL we have the following elementary modalities $ \Diamond = $ "eventually" $ \square = $ "always" I am aware the following distributivity laws hold i) $ \Diamond (a \lor b) = \Diamond a \...
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1answer
179 views

Personal question about Bernoulli's Diminishing Marginal Utility in money

I'm working for a multi-part question which is confusing me. Below are the questions and what I've tried so far. As well as the questions that I'm having difficulty with understanding. According to ...
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1answer
56 views

What sort of language does the language of second-order arithmetic become if the 'numbers' are the finite ordinals?

The language of second-order arithmetic is defined as follows (the wording of this definition is due to Henry Towsner from a pdf file of "[Chapter 4], "Second Order Arithmetic and Reverse Mathematics" ...
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1answer
72 views

Landscape of probability theory [closed]

I'm an engineering student who has taken one undergraduate course in probability theory, but that's all my exposure so far. I'm trying to get into machine learning and need to develop more of a ...
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3answers
1k views

Consistency of ZFC and proof by contradiction

I will start off by saying that I am an elementary student of mathematics and do not possess the deep and rigorous knowledge of most members of this site. Nonetheless, whilst learning how to do a ...
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0answers
100 views

better conjecture than the probabilistic model for primes

Are there some simple, unifying and convincing models about the properties we expect for the prime-numbers : that would be much stronger than the usual probabilistic model $\mathcal{P}(n {\...
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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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2answers
431 views

Are all questions solvable?

This is math/philosophical question. Are all problems solvable? By solution, I also mean that if a problem has no solution, then that is still a solution. What I mean is that for every problem, is ...
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1answer
137 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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3answers
893 views

How do we prove a set axioms never lead to a contradiction?

How can we be sure that a set of axioms will never lead to a contradiction? If there's a contradiction, we will find it first or later. But if there's no one, how can we be sure we choosen reasonably ...
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1answer
38 views

What makes Euclidian space univalent and topological space multivalent?

Here is a quote from wikipedia: Euclidean axioms leave no freedom, they determine uniquely all geometric properties of the space. More exactly: all three-dimensional Euclidean spaces are ...
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1answer
261 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
153 views

Why is the sigmoid function always written the way it is?

This might be a stupid question, but whenever I encounter the sigmoid function, it is written like this: Is there any particular reason why it is (it seems to me) never mentioned in the form: This ...
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0answers
110 views

Is axiomatic method the only way to introduce a mathematical theory?

I have some course in mathematics. Group theory, Ring theory, topology and etc. All of this theory begin with axioms. Whether every theory in mathematics should getting started with axioms? Is ...
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1answer
99 views

Is there a theory in physcics that has not a mathematical theory?

On person said to me that there is some theory in physics that they have not a good formalization yet, i.e. there is not a mathematical theory that covered them. Is this true? If yes, then without a ...
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1answer
217 views

Did some ultra-finitists suggest which number should be the largest?

I came across the ultra-finitism, the idea that there is a "largest number". Even most ultra-finitists admit that the "largest number" cannot be exactly defined. Therefore my question : Did some ...
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1answer
49 views

the way we do it or the way it should be done [closed]

in the preface to one of his works Sir Bertrand Russell writes: .. in mathematics the greatest degree of self-evidence is usually not to be found quite at the beginning, but at some later point .. ...
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3answers
208 views

weak counterexamples and the law of trichotomy in intuitionism

I've been reading a reaserch in Stanford's website about Intuitionism and I can't really understand what is a weak counter example and how does the intuitionistic continuum lookes like and it's ...
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1answer
394 views

In what way intuitionism is unique in the constructive approach [closed]

I am writing a paper on the subjects of constructivism and intuitionism. While I do know that intuitionism is a part of constructivism; it is also written that a lot of logic in intuitionism is unique ...
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1answer
162 views

Why isn't there an universal standard for mathematics? [closed]

I've been studying mathematics (e.g., calculus, linear algebra, statistics, etc) as part of my curriculum in informatics or computer science (whatever you want to call it), so I'm accustomed to ...
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0answers
99 views

Not computable (but definable) real numbers, not related to the halting problem?

The existence of not computable real numbers has been eating at me for a long time. However, I have not seen any example of such a number not related to the halting problem. Could any such number ...
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7answers
5k views

Some confusion about what a function “really is”.

Despite my username, my background is mostly in functional analysis where (at least to my understanding), a function $f$ is considered as a mathematical object in its own right distinctly different ...
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2answers
278 views

Fair coin toss probability question

Can anyone help me with this probability question?? You are out to dinner with two friends. You discover that there is only one remaining slice of chocolate cake, and so you all want to devise a fair ...
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2answers
1k views

Why do we find Gödel's Incompleteness Theorem surprising?

Gödel's First and Second Incompleteness Theorems are well-known, and usually taught by most colleges in undergrad logic courses. In my logic course I'm taking, we went over the proof of Gödel's ...
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3answers
282 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 ...
2
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1answer
534 views

Difference between Deduction and Induction

I would like to know what is the difference between deduction and induction. Mathematical induction I know well, but now I would like to look at these from a philosophical point of view. All help is ...
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3answers
722 views

Adding zero, multiplying times one… are they mathematical operations?

I saw a mathematician explain how the number 1 is not considered a prime number despite it fitting the traditional definition for a prime number; it is a natural number that can be divided by 1 and by ...
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0answers
61 views

What are the implications of exponential growth of information?

The fact that linear growth of information (1,2,3.. bits) results in an exponential growth of the accumulated information (2,4,8.. possible values) seems to be very fundamental. What are the ...
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2answers
292 views

$\sqrt{-1}$ is both a positive and a negative number [closed]

I contend that there is a third category of number (in addition to positive and negative numbers), which are neutral. For the sake of expression, let us call these numbers neutral numbers. Zero, for ...
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5answers
3k views

Why isn't finitism nonsense?

This is a by product of this recent question, where the concept of ultrafinitism came up. I was under the impression that finitism was just "some ancient philosophical movement" in mathematics, only ...
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3answers
166 views

Does one proof imply the existence of other proofs?

I asked my math professor if the existence of one proof for something implies the existence of other proofs we may or may not have found yet (she didn't know). Another way to phrase it: are there ...
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2answers
82 views

Infinity is Many-One

Bertrand Russell in Introduction to mathematical philosophy states, "It will be observed that zero and infinity, alone among ratios, are not one-one. Zero is one-many, and infinity is many-one." (P.40)...
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1answer
70 views

How many different mathematical problem classes are there?

Not a very mathematical question. Was just wondering... Can every math problem ever created, and that ever will be created be reformulated into a particular example of a finite set of problem ...
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3answers
1k views

What exactly does tautology mean?

I'm struggling to understand the actual meaning of tautology. I know how to determine tautology with a truth table. But I don't understand what it actually means, so consider the following reasoning: ...