How can you make sense of “equinumerosity” in Hume’s Principle in a logicist approach to math, without first having functions defined?

What is logicism in philosophy of mathematics?

In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of ‘logic’ — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic.

Is hume’s principle analytic?

For Frege, a statement is analytic, roughly, if it is provable from general logical laws and admissible definitions. The latter charactarisation is similar to the former, but the ways in which it differs suggest that the latter cannot be adopted if Hume’s Principle is to be deemed analytic.

What is a number Frege?

Frege’s definition of a number

Frege defines numbers as extensions of concepts. ‘The number of F’s’ is defined as the extension of the concept G is a concept that is equinumerous to F. The concept in question leads to an equivalence class of all concepts that have the number of F (including F).

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What is neo logicism?

by Zermelo-Fraenkel set theory (ZF). Mathematics, on this view, is just applied set theory. Recently, ‘neologicism’ has emerged, claiming to be a successor to the original project. It was shown to be (relatively) consistent this time and is claimed to be based on logic, or at least logic with analytic truths added.

Which philosophy is represented by aristotelianism?

In metaphysics, or the theory of the ultimate nature of reality, Aristotelianism involves belief in the primacy of the individual in the realm of existence; in the applicability to reality of a certain set of explanatory concepts (e.g., 10 categories; genus-species-individual, matter-form, potentiality-actuality, …

How do you know what is right according to intuitionism?

And the third belief of intuitionism is that human beings are able to know these truths through intuition. Now, this theory admits that we can twist and misinterpret basic truths because we are emotional beings, but the fact remains, intuitively, we know if something is right or wrong.

Does Math reduce to logic?

Ultimately, many believe that it is not possible to entirely reduce all of mathematics to logic, given Gödel and Tarski’s results. Some still do; however, as of this moment, no logicist program has done exactly what those like Frege and Peano wished it would do.

Is mathematics reducible to logic?

The basic claim of logicism is that mathematics is really a branch of logic. This is sometimes expressed by saying that mathematics (in this case, arithmetic) is reducible to logic.

What does axiom mean in math?

In mathematics or logic, an axiom is an unprovable rule or first principle accepted as true because it is self-evident or particularly useful. “Nothing can both be and not be at the same time and in the same respect” is an example of an axiom.

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What might be an example of an axiom in geometry?

Examples of axioms can be 2+2=4, 3 x 3=4 etc. In geometry, we have a similar statement that a line can extend to infinity. This is an Axiom because you do not need a proof to state its truth as it is evident in itself.

What is the importance of axioms describe what happens if it is missing?

Axioms are important to get right, because all of mathematics rests on them. If there are too few axioms, you can prove very little and mathematics would not be very interesting. If there are too many axioms, you can prove almost anything, and mathematics would also not be interesting.

What is axiom used for?

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning ‘that which is thought worthy or fit’ or ‘that which commends itself as evident’.

Is math always true?

Math is completely correct. The problem is that we set the rules for math and those rules do not always follow nature. As a matter of fact they are not even close as all the natural systems have limits and boundaries.

How do you prove math theorems?

Quote from the video:
Youtube quote: In other words they are congruent. Well one way to do that is to write a proof that shows that all three sides of one triangle are congruent to all three sides of the other triangle.

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