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Here is partial preliminary answer: Logicism, intuitionism and formalism are three traditional views about the nature of mathematics. Formalism was introduced by the German mathematician David Hilbert, and it holds that all mathematics can be reduced to rules for manipulating formulas without any reference to the meanings of the formulas. Thus, according ...


6

The short answer is that Gödel's incompleteness theorems are not contradictory, and arguably they are not paradoxical either, except in so far as they upset our preconceptions about provability and axiomatizability. They do express important limitations on what can be proved in formal systems. A longer answer is that Gödel's incompleteness theorems are ...


5

Your question is grammatically sound but it sort of defeats itself by suggesting to rewrite definitions but then judge them by the old definition. I'm not accusing you of any illegal substance abuse, but this is what I call a "stoner question". My favorite real-life stoner question that I've been asked is "what if clocks were like ants?" ...


5

There is a lot of writing both in favor and against AC from a philosophical standpoint - e.g. in favor see Penelope Maddy's Believing the axioms. However, there are also more mundane issues. I think that, whether or not it's ideal, a key point here is usability. An answer like this may seem dubiously appropriate at philosophy.stackexchange, but I think it'...


5

Newton, with the bias of his time, sought to formulate all his proofs in geometrical terms. He was also a freemason, a cult-tradition that identifies divinity with architecture, and geometry - and that tradition of divine geometry goes back at least to Pythagoras. Newton had to do a great deal of work to recast things in that way. Was it a waste of time? It ...


5

Computability of real numbers from Turing machine or Church's lambda calculus isn't necessary for a generic existence. According to computable number reference here: Every computable number is definable, but not vice versa. There are many definable, noncomputable real numbers, including: 1.any number that encodes the solution of the halting problem (or any ...


4

I would respond that the question is moot: it is too difficult (at present) to talk in a meaningful way about the probability of a mathematical statement. This is because one of the standard interpretations (or classes of interpretations) of probability is as quantifying over "possible states" - so that when we say that flipping a coin results in ...


4

None of the other answers (so far) addressed the actual mathematical inquiry here. Firstly, your question is straightforwardly formalized as follows: Suppose you have an oracle O that when given any computable formal system S and a sentence Q can determine in finitely many steps whether or not Q is a theorem of S. Then you ask whether or not O can be used to ...


3

Descartes considered that he needed Euclid's parallel postulate: Descartes identified space and the extension of matter, so geometry was, for him, about real physical space. But *geometric space, for Descartes, had to be Euclidean. This is because the theory of parallel lines is crucial for Descartes' analytic geometry - not for Cartesian ...


3

You ask, "To support its formal existence, what concrete definitions have been provided by mathematicians/logicians?" The real numbers can be defined in several equivalent ways. One way is to start with the set of infinite sequences of rational numbers. Some of these sequences are Cauchy sequences, which means that the elements get arbitrarily ...


3

I am not entirely certain what exactly you mean, but here are some thoughts which might qualify as an answer. The enterprise of mathematics makes use of both discovery of unexpected results and invention of new branches of mathematics. To be all-knowing, your ultimate mathematician-being would need to have in its possession every as-yet uninvented, new ...


3

I'm not sure exactly what you mean by "discovery" here. If your Perfect Mathematician knows every true math statement, does it mean he knows every theory possible to define within this mathematical system? You know there is an infinite number of those? If so, your PM must be some kind of demon, all right. But doesn't his posession of god-like ...


3

Short Answer It sounds you're struggling to understand the relationship between three fundamental theories. Naive set theory is the theory used historically by Gottlob Frege to show that all mathematics reduces to logic. Type theory was proposed and developed by Bertrand Russell and others to put a restriction on set theory to avoid Russell's paradox, and ...


2

No, they do not. First, even if picture proofs were empirical it does not mean that it can not be derived by other, non-picture, means. Just because we can surmise 1+1=2 from our experience with common objects does not mean that it is empirical either. "An a priori science is one whose knowable truths are all knowable in an a priori way, allowing that some ...


2

You are confusing two uses of the word argument. In one sense, an argument is an extended discourse with limited aims such as education or persuasion. In the second sense, argument is a synonym for the technical term inference which is the process by which a single proposition can be constructed from a collection of premises (sometimes unstated). So, in ...


2

The basic confusion in the submitter's argument is a misunderstanding of what is meant in the quote by 'the next theorem'. One must distinguish between those theorems which may be proved in principle, which is what the submitter's argument refers to, and those theorems which have actually been proven (and physically written down!) in fact, which is what the ...


2

SHORT ANSWER If you ask it, and you get a meaningful answer, then it is meaningful. This is an operational definition of semantics and is utilized by the Turing Test. The validity of operational definitions is based on correlation to establish dependence. Idealist and realist positions on the origins of math can occur in a nuanced way, depending on your ...


2

Numbers are means for quantifying reality. This quantification requires counting, where one phenomenon is separated from another or united: In counting 2 oranges we unite them into one set in one respect while observing that 1 orange exists in a variety of states as 2 oranges in a seperate respect. Numbers do not exist without counting, and the most basic ...


2

There's a lot to unpack here, so first I'm going to summarize my understanding of your question: You disagree with Platonism. Based on what you have written, you sound like a nominalist (i.e. you think mathematical entities are representations of "real" or physical entities). Some people who support Platonism make an argument of the form "If aliens (or ...


2

According to the empirical research of the Natural Semantic Metalanguage project, there are a group of around 65 "semantic primes". These are core concepts shared by all human languages, which are the basis of all other meanings, and which cannot be usefully broken down into other concepts; any definitions of these primes will inevitably end up more ...


2

This answer will consider ways that zero and one need to be described in terms of each other using the axioms of Peano arithmetic. Wikipedia describes a model of the axioms of Peano arithmetic as a triple: A model of the Peano axioms is a triple (N, 0, S), where N is a (necessarily infinite) set, 0 ∈ N and S : N → N satisfies the axioms above. The ...


2

I would like to paraphrase what you wrote: My limited understanding of set theory is that the mathematical objects described in sets exist in their entirety at any given instant in time. Quantum theory has pretty conclusively shown that even the most fundamental of particles still have a wave nature. things on a wave nature don't exist at just one ...


2

I think the title and body questions are subtly different. Here I'm going to address the title question, which I'll paraphrase for clarity as: What sort of "mathematical truth" can a non-Platonist make sense of? I think this is less strange than it may first appear, since there is an existing parallel: "sharp" vs. "fuzzy" referents in natural language. ...


2

The debate is what is commonly referred to as the Foundational Crisis of the early XX century (https://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis). A well-known example of one of the issues from which this crisis stemmed is Russel's paradox in Frege's foundations for mathematics. Hilbert's position on this is well explained in https:...


2

I agree with Noah Schweber's answer, but I think if we swap out Con(ZFC) for something else, we might still be able to find a plausible example of a mathematical conjecture for which we might say there is evidence for its truth even though we lack a proof. I have in mind the conjecture from computational complexity theory: P is not identical with NP. In ...


2

This is a misconception, you don't need to be able to enumerate the elements of a set. In naive set theory (which has its problems but is useful to explain the set concept here), a set is defined by some property. Every object that satisfies the property is contained in the set. The only thing you can ask a set is whether some object is contained in it. If ...


2

Most mathematicians are happy to use ZFC set theory or one of it's equivalents. These set theories support the "normal" real numbers. There are, however, mathematicians such as the intuitionists and constructivists who might be said to be "suspicious" of ZFC set theory for reasons not unlike what those you have expressed in your question. ...


2

According to mathematical nominalism, "existence" is reserved for things that exist physically. In this view, neither computable numbers nor uncomputable numbers exist. Mathematics can be sensibly viewed as a what-if scenario: what-if objects satisfying <certain definitions> existed? What would follow from that counterfactual hypothetical?...


2

Long comment When in mathematics we write 1+2+3+4=10 we are not making some sort of "metaphysical claim": we are asserting that when we evaluate the left-hand side expression (we "compute" 1+2+3+4) the process will terminate after a finite number of steps and the resulting value of the process will be the same as the right-hand side. Thus,...


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