Author Topic: Mathematical Musings (Read 793 times)

Holden · « **on:** August 16, 2021, 01:18:06 pm »

Hume's Principle:The principle that the number of things with the property F equal the number of things with the property G if and only if there is a one-to-one correspondence between those that are F and those that are G. ( As used by Frege).

Holden · « **Reply #1 on:** August 17, 2021, 12:46:12 pm »

When doing maths it is important to not take the idea of “truth” as something which is primitive in character.To assert “not Q” would be to prove that assuming “Q” leads to contradiction.

Holden · « **Reply #2 on:** August 17, 2021, 01:10:22 pm »

Principium Tertii Exclusi ,essentially ,boils down to the claim that any mathematical problem has a solution.

Holden · « **Reply #3 on:** August 18, 2021, 01:57:43 pm »

If one uses the Axiom of Choice, then, something very strange happens.It allows one to take a sphere in space, decompose it into a number of parts and then recombine the parts by making use of rotations into two sphere ,both of which have the same volume as the original one.

Holden · « **Reply #4 on:** August 20, 2021, 02:40:12 pm »

The algorithm which will find 1 in a binary sequence in which it is not possible that all terms are 0 will go on with the process of jotting down the sequence until we find 1. However, it might not provide the assurance that it would be found before the extinction of mankind or before the Sun goes nova.

Nation of One · « **Reply #5 on:** August 20, 2021, 07:32:00 pm »

Quote from: Holden

The algorithm which will find 1 in a binary sequence in which it is not possible that all terms are 0 will go on with the process of jotting down the sequence until we find 1. However, it might not provide the assurance that it would be found before the extinction of mankind or before the Sun goes nova.

I think you definitely possess the potential to "grok" Wildberger's theory; that is, you get the gist of his main arguments.

The field of decimal numbers

When the term decimal number is used in this book [Divine Proportions], it refers to a number such as π^2 /16 ≈ 0.616 850 275 . . . with a decimal expansion that is specified by an algorithm, computer program or function. All the constants of mathematics, such as √2, π, e, γ and so on, are decimal numbers in this sense, as are any arithmetical expressions formed by them, even those using infinite sequences and series, provided of course that these sequences and series are specified in a finite way.

Arithmetic with decimal numbers is thus intimately connected with the theory of computation. Unfortunately it is difficult in practice, and perhaps impossible in theory, to consistently determine when two algorithms generate the same decimal number.

This is a very serious deficiency, and implies that there is no effective notion of equality in the theory. So there is no general procedure to tell whether a given arithmetical statement involving decimal numbers is correct.

The topic of decimal numbers is a source of considerable confusion and ambiguity in mathematics. A proper development of the subject, which might have been a major agenda item for twentieth century mathematics, has yet to be taken sufficiently seriously. It is too difficult to be attempted here.

As an illustration of the problems that arise, it is generally accepted lore that the decimal numbers are countable and not complete. However with a careful examination of the definitions involved, it turns out that the reverse is true – the field of decimal numbers is both complete and not countable!

Despite such confusions, it is convenient to refer to numbers such as √2, π, e and so on, temporarily putting aside the conceptual difficulties. So the decimal number field will be flagged as an ‘informal’ field, awaiting a proper treatment.

The field of ‘real numbers’

Even more logically unsound than the field of decimal numbers is the field of so-called ‘real numbers’. Believers in ‘real numbers’, which currently includes the majority of mathematicians, assert that the decimal numbers of the previous section should be called computable decimal numbers, to be distinguished from ‘non-computable decimal numbers’. These latter ‘numbers’, which play the role of Leprechauns in modern mathematics, supposedly have infinite decimal expansions which are not determined by an algorithm, formula or computer program. In other words, these are decimal expansions which by definition cannot be described explicitly by finite beings such as ourselves, or our computers.

Holden · « **Reply #6 on:** August 21, 2021, 06:04:40 pm »

You are like the Sun that keeps Pluto from flying off into the outer darkness.I,obviously,am Pluto.

Holden · « **Reply #7 on:** August 26, 2021, 09:32:52 am »

Even if the collections in question are infinitely large, still , the number of recombinations is always strictly larger than the number of objects in the original collection.

Holden · « **Reply #8 on:** August 27, 2021, 01:02:28 pm »

Inconsistency creates trouble in logic because it leads to explosion.From a contradiction anything at all can be proved.

Holden · « **Reply #9 on:** August 28, 2021, 02:36:09 pm »

There are areas and situations which are really intractably inconsistent. The remedy is paraconsistent logic.

{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

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Author Topic: Mathematical Musings (Read 793 times)

Holden

Mathematical Musings

Holden

Primitive Truth

Holden

Principium Tertii Exclusi

Holden

Banach-Tarski Paradox

Holden

Algorithm

Nation of One

Re: Mathematical Musings

Holden

Re: Mathematical Musings

Holden

Cantor

Holden

Logical Explosion

Holden

Intractable