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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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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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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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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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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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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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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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 {\... 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 ... 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 ... 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 ... 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 ... 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 ... 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; ... 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 ... 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 ... 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 ... 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 ... 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 .. ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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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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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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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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)...