This is an audio version of the Wikipedia Article:
[ Ссылка ]
00:02:10 1 Deductivism
00:03:19 2 Hilbert's formalism
00:06:46 3 Axiomatic systems
00:07:09 4 iPrincipia Mathematica/i
00:07:52 5 Criticisms of formalism
00:09:35 6 See also
Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.
Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain
Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.
Listen on Google Assistant through Extra Audio:
[ Ссылка ]
Other Wikipedia audio articles at:
[ Ссылка ]
Upload your own Wikipedia articles through:
[ Ссылка ]
Speaking Rate: 0.9779106838448671
Voice name: en-AU-Wavenet-D
"I cannot teach anybody anything, I can only make them think."
- Socrates
SUMMARY
=======
In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be considered to be statements about the consequences of certain string manipulation rules.
For example, Euclidean geometry can be considered a game whose play consists in moving around certain strings of symbols called axioms according to a set of rules called "rules of inference" to generate new strings. In playing this game one can "prove" that the Pythagorean theorem is valid because the string representing the Pythagorean theorem can be constructed using only the stated rules.
According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other contensive subject matter — in fact, they aren't "about" anything at all. They are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics).
Formalism is associated with rigorous method. In common use, a formalism means the out-turn of the effort towards formalisation of a given limited area. In other words, matters can be formally discussed once captured in a formal system, or commonly enough within something formalisable with claims to be one. Complete formalisation is in the domain of computer science.
Formalism stresses axiomatic proofs using theorems, specifically associated with David Hilbert. A formalist is an individual who belongs to the school of formalism, which is a certain mathematical-philosophical doctrine descending from Hilbert.
Formalists are relatively tolerant and inviting to new approaches to logic, non-standard number systems, new set theories, etc. The more games they study, the better. However, in all three of these examples, motivation is drawn from existing mathematical or philosophical concerns. The "games" are usually not arbitrary.
Recently, some formalist mathematicians have proposed that all formal mathematical knowledge should be systematically encoded in computer-readable formats, in order to facilitate automated proof checking of mathematical proofs and the use of interactive theorem proving in the development of mathematical theories and computer software. Because of their close connection with computer science, this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition.
