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Language, Logic and Understanding

 

We make many decisions during the course of the day. Sometimes these decisions are guided by emotion, sometimes we just rely on a hunch, sometimes we rely on experience, and sometimes we analyze a situation logically and make a decision according to this logical analysis. But very few things in life are easy to analyze in a completely logical way; in most cases, our actual decisions are based on a combination of emotion, experience, and a little bit of logic.

 

However, when we want a conclusion that isn’t based on any emotion, or hunch, we want a conclusion that is arrived at purely by means of logical argument. This site is devoted to showing how many results that are commonly accepted as being the result of a completely logical argument are in fact flawed because of the failure to acknowledge the significance of the way language is used in the argument - a seemingly innocuous statement can contain subtle errors which render the statement illogical. Unless every aspect of a statement is very carefully analyzed with regard to the use of language by the statement, an ostensibly logical statement may actually contain subtle errors of logic. Even fairly innocuous looking statements can be difficult to analyze, see Natural Language and Reality.

 

Intuition

This site explains how intuitive errors may occur; in most cases it is because insufficient attention has been given to the use of language. If you are visiting this site for the first time, I suggest these pages as suitable starting points:

In principle, a logical argument should never rely on an unstated intuitive assumption. It is well known that intuition can lead to erroneous results, and that there are many examples of this having happened. So it should be the case that every logical argument should be carefully examined to ensure that it contains no intuitive assumptions. But there seems to be a blind spot when it comes to the possibility that the way that language is used in an argument might affect the validity of the argument. This possibility is commonly dismissed without any justification for its dismissal. But everything that is referred to by a logical argument must be referred to by symbols that belong to some language. And since that is the case, the fact that those symbols belong to some language is an inherent part of the argument, and is not something that can simply be ignored.

 

Self-reference

Much of this website deals with the confusion that occurs when levels of language are not clearly delineated. Kurt Gödel set the ball rolling on this in 1931 with his incompleteness theorem which hides its language confusion under an impressive looking facade of complexity. Amazingly, it has long being accepted as correct even though Gödel never actually proved the crucial step in his proof, and although his proof leads to a blatant contradiction, see Gödel’s contradiction. And over the years since that there seems to be an alarming increase in the willingness of certain academics to forgo the need for clear precise logical Picture: Snake eating itselfproofs of any claim, and now there are numerous people who like to call themselves “logicians”, but who are content to simply make a crucial assumption rather than actually make an attempt to prove it, and proceed to base an entire structure of claims based on that assumption. That assumption is that a completely formal language can actually reference itself - that is, that within a completely formal language there can be a sentence that explicitly refers to that entire sentence itself.

 

Despite their self-appellation as “logicians”, that isn’t logic, and the inane results of these assumptions aren’t logical - they are worthless. For an example of this sort of nonsense, see Halbach and Zhang: Yablo without Gödel.

 

Opinionated?

Most of this site is, naturally enough, based on logical and factual analysis. To provide some contrast, I decided to include some viewpoint based material here - this is where I get an opportunity to voice my opinion on various matters. Feel free to disagree.

 

 

30 Sep 2019    A Failure to Understand

I came across a website written by a Charles Fisher Cooper who writes articles on various subjects, including mathematics. In the section “George Without the Dots ” on his page Cantor's Diagonal Argument he claims he has a rigorous diagonal proof that overcomes objections to the standard presentation of the proof.

 

His argument proceeds by assuming that there is a set of ordered pairs, Picture: Dicewhere the first element of each ordered pair is a natural number, and the second is a decimal expansion of a real number where the digits of such real numbers are “randomly selected ”. The author asserts that each such “ordered pair has a counting number for the first number and a real number for the second.” He also asserts that no two such ordered pairs have an identical real number.

 

He then claims that what he has done is that he has “defined an effective method for generating unique real numbers, each of which can be assigned to a unique counting number.”

 

He then proceeds to claim that he can define another real number whose digits are different from every real number Picture: Infinityin the set of ordered pairs that he has described. He states that this new number is given by taking the ith digit of the ith such number and creating a new number by making its ith digit different to the ith digit of the ith such number.

 

He concludes that there will always be an extra number not in the original list. So you will always have more real numbers than counting numbers.”

 

The fallacies in the argument

One obvious fallacy is when he claims that he has “defined an effective method for generating unique real numbers, each of which can be assigned to a unique counting number.”

 

The fact is that he has not done so. According to his definition, the digits are random, therefore there is no information as to what the first generated real number is, and so you can't assign the natural number 1 to it. In fact you cannot know what any such real number is, and you cannot know even one of the digits of any such number.

 

It’s hard to know whether to laugh or despair when one sees arguments like this. Cooper has managed to assume that it is possible to define a one-to-one correspondence of natural numbers to all real numbers within a given mathematical language - which is precisely what Cantor’s diagonal argument proves is impossible. (Footnote: In fact there are fully formal proofs of this, see Fully Formal proofs of the Diagonal proof.)

 

The irony here is that Cooper’s Platonist assumption that random digits ‘exist’ independently of any unambiguous definition actually demonstrates the fallacy of such Platonist beliefs.

 

As Cooper himself almost manages to point out, there cannot be any real number Picture: Binary Numbersthat is not included in his description, since all possible sequences of digits are included in his description. So if there ‘exists’ such a Platonist set of such ordered pairs, there would be an enumeration of all real numbers, and so there would be a contradiction - since there must also ‘exist’ a Platonist diagonal number which is not any of the numbers in the set of ordered pairs, yet every possible sequence of digits already ‘exists’ somewhere in the set of all the ordered pairs.

 

The notion that there somehow ‘exist’ numbers that have “randomly selected ” digits is a Platonist notion that is lacking in any logical foundation and which leads to contradictions - the random digits are selected by whom? … or what? … some deity perhaps? It’s philosophical nonsense.

 

See also the post Random Numbers.

 

For more on Platonism, see Platonism, The Myths of Platonism, Platonism’s Logical Blunder, Numbers, chairs and unicorns and the posts Moderate Platonism and Descartes’ Platonism.

 

Note

While Cooper remarks that his web-page is essentially that which can be read in the book Metalogic: An Introduction to the Metatheory of Standard First Order Logic by Geoffrey Hunter, University of California Press (1973), there does not appear to be any part of the book that corresponds to Cooper’s argument that is discussed above.

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Footnotes:

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Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked. Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - any address will do, it does not require verification. Your e-mail will only be used to notify you of replies to your comments - it will never be used for any other purpose and will not be displayed. If you cannot see any comments below, see Why isn’t the comment box loading?.

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The Lighter Side

NEWS

Paper on the diagonal proof

There is now a paper that deals with the matter of language and the diagonal proof, see On Considerations of Language in the Diagonal Proof.

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Other recently added pages

The Myths of Platonism

 

Goodman’s Paradox

 

The Platonist Rod paradox

 

The Balls in the Urn Paradox

 

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Lebesgue Measure

There is now a new page on a contradiction in Lebesgue measure theory.

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Easy Footnotes

I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).

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O’Connor’s “computer checked” proof

I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.

Previous Posts

Random selection of numbers

Moderate Platonism

Descartes’ Platonism

The duplicity of Mark Chu-Carroll

A John Searle Inanity

Man versus Machine

Fake News and Fake Mathematics

Ned Block’s Blockhead

Are we alone in the Universe?

Good Math, Bad Math?

Bishops Dancing with Pixies?

Artificial Intelligence

Cranks and Crackpots

The Chinese Room

Links

For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem

 

– a page with general links:

Gödel Links

 

– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines

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Comments

Comments on this site are welcome, please see the comment section.

 

Please note that this web site, like any other is a collection of various statements. Not all of this web site is intended to be factual. Some of it is personal opinion or interpretation.

 

If you prefer to ask me directly about the material on this site, please send me an e-mail with your query, and I will attempt to reply promptly.

 

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Copyright © James R Meyer 2012 - 2018  
https://www.jamesrmeyer.com