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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'...


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 ...


4

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 ...


3

Why do we need algebra? Are you writing off Greek math as not math yet? Surely being able to leverage another sense for analogies is going to be a useful adjunct to insight. (And that sense is more kinesthetics than sight, there are good blind geometers.) The mathematicians of the Cartesian era added algebra to geometry, not the other way around. It ...


3

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

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

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

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 ...


1

Formalism would need a first-order logic. Wikipedia describes this as follows: First-order logic uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form "there exists x such that x is Socrates and x is a man" ...


1

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 ...


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 ...


1

I would say that mathematics is itself abstract and assumed to be consistent. Unless you subscribe to a form of solipsism I would argue that mathematical relations exist independent of your mind and body. We may discover the relations through observation of the physical world, and we may also use physical objects to demonstrate an abstract mathematical ...


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