Copyright © James R Meyer 2012 - 2018 https://www.jamesrmeyer.com
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.
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.
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 proofs 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 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.
Platonism is a belief that some or all concepts referred to by mathematical theories “exist” independently of any finite definition (see also the web-page Platonism). Gideon Rosen tells us that, in a chapter in the book “The Philosophy of Philip Kitcher” (Footnote: The Philosophy of Philip Kitcher, Ed. Mark Couch & Jessica Pfeifer, Oxford University Press, USA, 2016, ISBN: 9780199381357 The Philosophy of Philip Kitcher: Details) there is a book by himself and John Burgess, “A Subject with No Object” (Footnote: A Subject with No Object: Strategies for nominalistic interpretation of mathematics, Burgess, John P. & Gideon Rosen, Clarendon Press, 1997, ISBN: 9780198250128 A Subject with No Object: Details) that is a “full dress defense of moderate Platonism”.
You might be wondering what “moderate Platonism” might be, since you might imagine that either one believes that some concepts of mathematical theories “exist” independently of any finite definition, or else one doesn’t - you can’t believe something and simultaneously not believe it. Unlike the supposed entities of religion, Platonists don't ascribe supernatural powers to their supposed entities, so it’s difficult to fathom what a “moderate Platonist” is, and unfortunately you won’t find any definition of what it is in Burgess and Rosen’s book. And it makes one wonder what a fundamentalist hard-core Platonist might be by comparison. Burgess and Rosen are Platonists, and presumably think of themselves as “moderate Platonists”. They tell us that their book is a survey of anti-Platonist philosophies, but what you will actually find that the substance of their book is a collection of arguments that substitute logic by a selection of many well-known fallacies. (Footnote: See, for example, Dishonest tricks used in Arguments, A List of Fallacious Arguments, Fallacies - The Internet Encyclopedia of Philosophy, List of common fallacies at ‘No Beliefs’, Wikipedia entry - List of Fallacies.)
You don’t have to delve far in the book to find that it isn’t a balanced appraisal of anti-Platonist arguments, and to discover Burgess & Rosen’s inherent bias. In the preface of the book, they state:
“Nominalism (as understood in contemporary philosophy of mathematics) … is motivated largely by the difficulty of fitting orthodox mathematics into a general philosophical account of the nature of knowledge. The difficulty largely arises from the fact that the special, ‘abstract’ objects apparently assumed to exist by orthodox mathematics - numbers, functions, sets, and their ilk - are so very different from ordinary ‘concrete’ objects. Nominalism denies the existence of any such abstract objects. That is its negative side.”
Why is that a difficulty?
Since the very essence of the non-Platonist philosophy is that there is no need to assume the independent “existence” of any mathematical entities that correspond to mathematical concepts, why would that be “its negative side” ? One might as well assert that denying the existence of fairies has a negative aspect that should be deprecated by any rationally minded human.
Clearly, Burgess and Rosen don’t like non-Platonism, so it’s a negative for them - but its not a negative for non-Platonists; in fact, for non-Platonists, it is a positive aspect, since it eliminates extraneous ideas that are not essential for doing mathematics, and which often have detrimental and paradoxical consequences. Burgess and Rosen’s statements simply tell us about their own underlying beliefs rather than serving to elucidate.
Burgess and Rosen go on to assert that non-Platonists need to
“reinterpret accepted mathematics so as to purge it of even the appearance of reference to numbers, functions, sets, or the like”
That is utter nonsense. There is no reason why a non-Platonist would want to remove the mental concept of numbers, or the mental concept of functions, etc from mathematics. Burgess and Rosen’s claim is self-serving hogwash.
Further on in the preface they assert that their book provides an overview of the different arguments that non-Platonists use against Platonism. Well, if Burgess and Rosen think that the objective of a non-Platonist must be to “purge [mathematics] of even the appearance of reference to numbers, functions, sets, or the like”, then they are barking up the wrong tree, and constructing an utterly worthless straw man. They act as though as soon as a mathematician uses the words “exists”, or “function”, or “number”, that he must be referring to an actual existing non-physical thing that is independent of any definition, whereas he may only be referring to a definition given by a finite sequence of symbols of some language.
Burgess and Rosen act as though the onus is on non-Platonists to prove that mathematical concepts do not “exist” as some sort of non-physical thing independently of any finite definition. But the onus is on Platonists to prove, not only that they “exist” independently, but also to provide a rationale as to how we might have any way
given that there can be no physical interaction with such postulated “things”.
Burgess & Rosen continue:
“Actually, there is not one programme of nominalistic reinterpretation or reconstrual, but several, since nominalism is a loose cluster of positions, and different nominalists prefer quite different strategies and methods.”
Yes, that is what Burgess & Rosen like to assert, but the simple denial that mathematical entities have any existence independently of any finite definition is a very straightforward notion and defining a group of people who hold such tenets presents no difficulty whatsoever.
The fact is that the literature is replete with extremely poor arguments and vaguely stated assertions. It’s easy for a Christian to fill a book with feeble arguments against the existence of a god, but it’s a classic error of logic to conclude that that implies that there must be a god. In the same way, it’s easy for Burgess and Rosen to fill up their book with instances of extremely poor arguments against Platonism, (Footnote: Examples of such poor arguments are various arguments against the The Indispensability Argument. Indispensability arguments don’t have any clear definable core rationale, so to engage in detailed discussion regarding the vague assertions in such arguments inevitably results in verbose bullshit on both sides.) and in precisely the same way, it would be a classic error of logic to then conclude that Platonism is necessarily true. (Footnote: In any case, it is easy to show that mathematical Platonism is inherently contradictory, see for example The Courant & Robbins Contradiction, Lebesgue Measure, Sums of infinitely many fractions: 1, Sums of infinitely many fractions: 2, Assuming that the diagonal number “exists”.)
Further on in their introduction, while Burgess & Rosen glibly state that “a nominalist is one who denies that abstract entities exist”, they continually twist that straightforward statement by an incessant muddying of the waters. They continually imply that whenever a non-Platonist refers to an abstract concept, he must be in some way invoking the existence of an independent non-physical thing that “is” that abstract concept, whereas the reality is that for the non-Platonist, the abstract concept is simply a mental concept that has been encapsulated by a combination of words of some language.
Burgess & Rosen go on to say:
“We say, for instance, that Jane Austen wrote six novels; and in so saying we appear to imply the existence of the six novels that Jane Austen wrote. But just what are these novels? Austen wrote the novels by writing certain manuscripts. Yet the novels are not the author’s manuscripts. A novel, at least once it has been published, does not cease to exist if the original manuscript is destroyed. One reads the novels by reading certain hardcovers or paperbacks. Yet the novels are not the hardcovers or paperbacks. If one borrows a paperback of Emma, reads it, buys a hardcover, and reads that, then one has read not two novels, but one novel twice. The novels, whatever they may be, are not made of paper and ink. They are not ordinary things: they are not things of the sort philosophers call concrete.”
This is classic case of straw man construction - construct a scenario complete with various assumptions that are in accordance with the authors’ inherent bias, and present it as a reasoned argument. Each Austen novel is simply a sequence of words of English, and the presumption in any reference to that novel existing includes a presumption that at least one physical copy exists in some form. (Footnote: A film or play based on the novel, however faithfully done, is not the novel. And a translation is not the novel, although a reasonable imitation of the original might be created from it, that would not be the actual novel.) Austen’s novels are simply an instance of where there are multiple physical containers of a certain collection of information, where the actual physical form of the containers do not need to be identical. There is no need for the notion that there “exists” somewhere in some ethereal non-physical sense a “thing” that constitutes the information content of the novel.
Now, even if all copies - of any sort - of such a novel were destroyed, such that we could never reconstruct that novel, we could still refer to that novel, and say that Jane Austen wrote that novel, although we would not have the complete information that was contained in the novel. That novel would no longer exist in any form as a physical object - in the same way that, for example, a prehistoric creature no longer exists. But that does not mean that we could not talk about its prior existence; of course, we can refer to previously existing objects, and we may be able to state things about them. In neither case is there any reason to suppose that there is any sort of continuing non-physical independent existence of the novel or the creature. Similarly one could talk about Joe planning to write a novel, before he has written any of it. That does not mean that as soon as we refer to Joe’s novel in that context, that that novel somehow either already always existed in some non-physical sense, or that it has suddenly popped into existence in some non-physical sense. It will not exist until Joe has written it (if he does write it). See Non-physical existence below for another comment on Burgess & Rosen’s notions regarding written texts.
Burgess & Rosen continue with:
“We say, again, that six is the number of Austen’s novels, and that six is a perfect number. In so saying we appear to presuppose the existence of something, namely, the number six. But what is it? Surely not any particular inscription of the Roman ‘VI’ or the Arabic ‘6’: those numerals only denote the number. Surely not any specific sextet or hexad, either: they only exemplify it. Numbers are if anything further than novels from being ordinary things. They again are things of the sort philosophers call abstract.”
No! If I said “six is the number of Austen’s novels, and six is a perfect number”, I would not be presuming that the number “exists”. All I would be doing is applying a description to a group of novels, and making a remark about a property of the description. I could also say that Austen’s novels are 19th Century, and that the 19th Century is important for its place in human history, and by so doing I would not be making any presumption that there “exists” an independent non-physical entity that is the 19th Century. By using the term “six” I would simply be noting that certain groups of things have common properties.
When Burgess & Rosen say “those numerals only denote the number”, that includes the hidden Platonist assertion that the “number” “exists”. The reality is that a Roman “VI” is simply a symbol sequence of a particular language system. I could also note, for example, that “sei” represents the same concept in Italian. So what? Why should it be noteworthy that there are different symbol sequences in various languages that describe a particular property? In general we don’t think it strange that there are different words in different languages that refer to the same mental concept, so why should we think it strange for the concept of number? For example, I don’t think of “politics” as some sort of non-physical “existing” thing that “exists” independently of any reference to it in some language.
Of course, the concept of a number is an abstraction, where as much as possible of any real world association has been stripped away. But that can never conceal the fact that the concept of number arose out of real world observation of groups of objects. The concept of number did not suddenly arise when some philosopher had a transcendental vision and thought, “My goodness, there exist amazing non-physical things, and I have just had an astonishing connection to them, even though my body is physical and these things are not. I will now write down what I know about them.”
No, of course not. The concept of numbers developed slowly over time, from simple notches on a stick to more complex concepts. But until relatively recently, all such development was based on real world observation, and on what worked and what did not when applied to real world situations. Philosophical concerns about such matters as the “meaning” of numbers were utterly irrelevant to builders and traders who used the concept of number in their daily work - and they still are.
Further on, Burgess & Rosen say:
“Colloquial speech and commonsense thought are liberally sprinkled with assertions appearing to involve non-concrete entities of all sorts, from novels to numbers. The language and theories of the natural and social sciences are not sprinkled but saturated with assertions apparently involving those quintessentially abstract entities, numbers: the fine structure constant, the gross national product, and so on indefinitely. As for pure mathematics, every branch of the subject abounds in assertions explicitly labeled ‘existence theorems’, beginning with Euclid’s Theorem on the existence of infinitely many prime numbers. All this means that there is a wide range of widely accepted theories, including many commonsense, most scientific, and virtually all mathematical theories, to which you as an imagined convert to nominalism apparently cannot give any credence. And yet you cannot simply discard these theories, either.”
Yet again, we have an appeal by straw man construction according to the authors’ biases, heaping assumption upon assumption. By the authors’ assumption that any reference to abstraction implies a reference to a non-physical existing thing, that means that any communication with those authors by a non-Platonist would have to state, every time one refers to a mathematical concept, something like, “Actually, when I refer to a natural number, I am referring to the concept of a natural number as defined, for example, by the Peano axioms, and I am not in any way implying any independent non-physical existence of a natural number.”
We’re only a few pages into the book, still in the beginning of the introduction, but it is already quite clear that Burgess & Rosen have an aggressive Platonist agenda to promote, and that they will include any amount of irrelevant verbosity and fallacious argument in an attempt to support their cause. The old adage applies: If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck. And despite Burgess and Rosen’s claims that their book is an overview of anti-Platonism, it has all the hallmarks of unabashed pro-Platonist propaganda.
Suppose that John writes an article. A short time later, Jane, without having any knowledge of John’s article, writes, by sheer coincidence, exactly the same article - it was a very short article! John decides to print a few copies of his article and publish them, but after a short time he decides it is not good enough and he destroys all copies of his article, and no-one has taken a copy of it. Jane also publishes and continues to publish copies of her article. Now, according to Burgess & Rosen, John’s article was, at least at one point in time, a non-physical existing thing. And so is Jane’s article. And the articles themselves are exactly the same. But are they the same non-physical existing thing? To say that they are divorces the authorship from the article, but conceptually, we would consider that the current published article is Jane’s article, and that Jane has established her right to be considered the author of the article, and our mental concept of “Jane’s article” reflects that.
So, given the notion that there “exist” independently existing non-physical objects that accurately encapsulate our mental notions, then either there must be one non-physical thing that encapsulates the mental concept of both the article and its origin, or there is a non-physical thing that encapsulates the mental concept of the article, and another non-physical thing or things that encapsulate the mental concept of the origin of the article.
The former possibility gives rise to the difficulty that when I refer to the “square root of 6”, the corresponding non-physical thing somehow includes the information that I am the origin of it, which means that my “square root of 6” is a different thing to your “square root of 6”. And if they are different, then what might we mean if we were to assert that we are talking about the same thing?
The problem with the latter possibility is that it gives rise to infinitely many nested such things. For example, for every set of three things there would be both the number three thing and its origin thing. And for every set of three such origin things there would be both the number three thing and its origin thing. And so on, and so on. The notion that such things might “exist” is becoming more and more preposterous.
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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.
There is now a new page on a contradiction in Lebesgue measure theory.
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).
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.
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:
– and a page relating specifically to the Gödel mind-machine debate:
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