9

It think there's something confused in this question. Mathematical constants like pi are exact. Pi is the ratio between the circumference and diameter of a perfect circle on a flat plane, and the fact that we cannot calculate an exact value for it does not mean that the constant itself is not exact. Planck's constant might suggest that we cannot have a ...


5

You can address this issue through significant figures, which indicate the level of precision with which a value has been measured. If I say that I walked 2 miles, that's a rough estimate - I'd be justified in saying that if I had walked anywhere from 1.5 to 2.4 miles. If I say that I walked 2.000 miles, that is a much more precise number - I'd only be ...


5

First, let's observe (in response to a comment) that we can get infinitely nonequivalent sentences; this isn't what you want, but it demonstrates that the situation is nontrivial. Namely, we can talk about cardinality, via the following statements which are easily seen to be appropriately expressible: N(n): There exist x_1,...,x_n which are all distinct. ...


4

Throughout, I'm assuming that ZFC is consistent. There are a lot of confusing points here, and I don't really understand what you're setting up with bijections. However, I believe the key mistake you make isn't actually related to sizes of sets at all, but rather a serious misapplication of the incompleteness theorem. The language about infinite sets and ...


3

The answers to both "some kind of platonism" and "infinitely many universes instantiating mathematical structures" is no. Born is closer to Hegel than to Plato, and even further from Tegmark than from Plato. His views are described more systematically in the book Physics in My Generation (1966), the chapter Symbol and Reality: "In every field of ...


3

Unless one is considering a "free logic", classical first order logic assumes all models that would be permitted to provide countermodels have a non-empty domain. So there exists a member of the domain which one may name t. Let x be a variable representing any member of the domain. Here is a proof of the above using a Fitch-style proof checker: The first ...


2

I'm not sure about your approach but I'd agree that the issue of precision is an important philosophical topic that deserves more attention. I don't know of anyone who explores this as a philosophical issue. It is not just to do with Pi. No location on the number line can be identified with precision if all locations are infinitely divisible. EDIT: Doh! I ...


2

I'm going to kind of ad lib here, but as a formalist, hopefully at least something of interest comes out of this. For a formalist, Semantics is how mathematics gets its practical application. When I say something like "one plus one equals two", I am discussing something logical and syntactic, but when I say "one orange plus one orange equals two oranges", ...


2

From the comments: I tried to use Existential Elimination but I can't figure out how to do it properly. Existential Elimination: When given that an existential statement (eg ∃z P(z)) holds, and show that a statement (eg Q) may be derived when we assume a witness for the existential (eg P(c), where c is a term that does not occur within P(z) or Q), then we ...


2

Here is a forward proof that gets by without contradiction, constructed with the Natural deduction proof editor and checker: Your problem presumably was that you can't just eliminate the existential quantifiers directly to replace them by an individual constant -- this would be unsound, and an incorrect application of the ∃E rule -- but rather the rule ...


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

Here are some quotes from Max Tegmark's "The Multiverse Hierarchy" hopefully answering the OP's question: What exactly does he mean by self-aware substructure? Tegmark assumes that "all aspects" of reality are isomorphic to a mathematical structure. Let us now digest the idea that physical world (specifically, the Level III multiverse) is a mathematical ...


1

Let me try to capture what you're doing with a general comprehension (= set-formation) principle: (C) For any property p, there is a set [p] such that (i) everything with property p is in [p] and (ii) whenever X is any set containing everything with property p, we have X is a subset of [p]. Basically, [p] is the smallest set containing everything with ...


1

If Max Tegmark is using the words "self-aware substructure of the universe", then he is referring to agency and the beings who possess it, for example us. If the universe is perceived as a mathematical structure, than not all of the structure possess the capacity for perception, hence the delineation of a sub-structure which by definition means a smaller ...


1

Uncertainty and infinity are related but different concepts. On the one hand, uncertainty is a concept that expresses a state of modality, or whether or not knowledge or information is complete or certain. This is related to such concepts as probability and determinism. Infinity is a concept related to cardinality and ordinality, that is to say counting and ...


1

The matter of precision is handled philosophically in the discussion of the infinite. In mathematics, for instance, precision is handled by the concept of the limit. Limits are particularly important in the philosophy of mathematics in topics such as the philosophy of set theory or analysis. If you are genuinely interest in this topic, I would recommend ...


1

I think your proposition is actually reversed. Rather than mathematics being the culmination of analytic philosophy, analytic philosophy was an effort to rebuild philosophy on the already-proven ground of mathematics and the natural sciences. People like Frege, Russell, and (early) Wittgenstein leveraged mathematical methods to try to build a more ...


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

I have a different understanding of mathematics than the one visible in the interesting contribution https://philosophy.stackexchange.com/a/32859/40722. I will provide some reasons here. I disagree with the assumption that all humans will agree ultimately upon the same mathematical truths as there is no such thing like mathematical truth. There are, however,...


1

Wittgenstein's point is that every standard is at some level strictly conventional: we agree that it is so, explicitly or implicitly, and by agreeing make it so. We cannot talk about the measurement of the standard without falling into self-reference. When a father answers this question, he would (obviously) say that the meter bar is one meter long. But we ...


1

Wittgenstein states, "one can say neither that it is one metre long, nor that it is not one metre long". The first condition denies something while the second permits something. Each activity needs to be considered separately to make the sense that Wittgenstein had in mind. What was he denying? He was denying that the role of the standard metre is something ...


1

I'm actually happy this question popped up in the feed because I'm amazed no one has pointed out the historical changes of the term "science". Unfortunately I don't have the appropriate amount of time to write a full piece of answer, but I'll attempt to draw some general lines. If you consider "science", or moreover the scientific community, as dynamic, ...


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