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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'...


6

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 ...


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

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 ...


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 ...


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 shall suggest that you first make sure that you fully understand basic FOL (first-order logic) and the technical details and proofs of some of the crucial theorems about it up to and including the semantic completeness of FOL and the Godel-Rosser incompleteness theorems. The reason for this is that it is impossible to have a proper discussion about the ...


1

Contrary to some commenters here, there is a vast difference between mathematics and language, despite the fact that any sentence can obviously be translated into mathematized "information." Russell, the Logical Positivists, and others set out to rid language of its murky qualities by reducing both language and mathematics to logic. While the work ...


1

The mathematical language is simply a more rigorous way to talk about the world. There is no limitation to it in this respect that wouldn't be a limitation to any language. That nobody knows today how to express jokes, puns and poetry mathematically does not imply that they could not possibly be expressed mathematically. There was a time when nobody knew how ...


1

Short Answer Both disciplines use symbols in a truth-centered, rule-based meaningful way, but mathematics is built on logic and is more contextual and covers topics such as known and unknown quantities, length, area, volume, direction and position, and shapes and their transformations. For instance, even simple arithmetic tends to be "built" on ...


1

Mathematical logic is a species of symbolic logic, itself a species of formal logic, which started essentially with Aristotle's syllogistic 2,500 years ago. Formal logic has always been understood by logicians as an attempt to represent or model human deductive thought. Mathematical logic, too, was initially an attempt to model what Boole called the "...


1

Classical arithmetics are based on multiple assumptions regarding human perception. 1+1 is not necessarily 2 in other contexts. A simple example: 1+1=2 only if you assume that the objects you are adding are isolated systems. When you make the operation 1+1 and the systems interact, results different than classical addition can be expected. 1+1 points, added ...


1

It seems what you are referring to is what is usually known as the computational theory of the mind, which is the ontological framework behind cognitive science and their most precious toy, artificial intelligence. The answer to your question might be in this article: https://plato.stanford.edu/entries/computational-mind/ The computational theory of the ...


1

This is certainly an issue: whereas the consistency of weak axiom systems (like no axioms at all, just the inference rules of first-order logic) can be demonstrated in a very strong sense (basically, if we can exhibit a model and verify that those axioms are satisfied in it in an "appropriately constructive" way), this isn't satisfactory at all since per ...


1

for every conception of geometry there is an associated system of algebra which captures the mathematical truths contained in that geometry. as such, the two fields are "joined at the hip", and mathematicians can freely switch back and forth between the conception of geometry as lines and points in space, etc. and the conception of (for example) systems of ...


1

It is WE who began the process of counting : A process which generates numbers. (We defined 1 as: single instance of counting, 2 as: an instance of counting and again an instance of counting, and then formalized them). If these abstract numbers cannot exert influence on world on their own (which I am sure they don't), does it even makes sense to argue of ...


1

It seems to me that the way you explain the formulation of numbers is off-target. The concept of 'countability' rests on two basic cognitive orientations: That the universe (or perceptual world) is primarily composed of discrete, independent objects. That these discrete, independent objects can be arranged into 'kinds' (or 'categories', or 'classes') ...


1

Why would someone bring in an altogether different notion of their abstract reality, when the argument about numbers being a projection of reasoning (mental faculty/intelligence) is the most logical explanation? There is no such a thing as "the most logical explanation". An explanation is logical or it is not (and, most often, it is not). So, I will guess ...


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