A typical discussion about my work
Many persons who come across this site manage to convince themselves that I must be wrong, without actually applying any logical thought process in order to arrive at this conclusion. Instead they rely on pronouncements by various authority figures, and simply assume that those authorities must be right. When pressured to actually attempt to provide a logical argument to back up their convictions, they inevitably demonstrate a lack of ability in logical argumentation, and instead engage in long-winded irrelevancies, and demonstrate their lack of in-depth knowledge of the subject matter. It might be said that such people exhibit the Dunning–Kruger effect. (Footnote: The Dunning–Kruger effect is a description of a condition where people have an illusion of their superior ability, whereas they are in fact unable to correctly assess their own ability, and assess their cognitive ability as greater than it is. See Kruger and Dunning’s paper Unskilled and Unaware of it … Inflated Self-Assessments.)
I have copied below the contents of a fairly typical discussion of this sort that began on the comments section of this site. I started this page because the commenter was writing longer and longer comments which were becoming too unwieldy for the home page. This person says that he has been in contact with me previously and I have a good idea as to his identity, but apparently he does not want anyone to know who he really is and he hides behind the pseudonym “Nova Kant”. This ploy is often used by commenters here - why are they afraid to reveal their true identity?
After much prodding of this person to actually try to find an example of an error in my work, he said that I had made incorrect statements about a formula in a book. As the discussion continued, it became obvious that he was talking about how he wanted to describe the formula, not how it was actually described in the book, and that he had not taken the time to become cognizant with the book, nor the details of the formula in the book, nor with my article on it. When I pointed this out he resorted to making excuses and left.
Maybe at this point you should start thinking there is something you do not understand and try to learn what it is, that absolutely everyone else seems to understand? Maybe?
James R Meyer
All anyone has to do is to point out the errors on this site. There’s well over a hundred pages on the site, so if I’m not understanding what I’m talking about, there shouldn’t be any difficulty.
Hyperbole such as yours is always suspect - absolutely everyone else? really? In fact you only have to look at various comments on other pages on this site to see where I have had to correct many misunderstandings of basic concepts.
Your vague comment is typical of many others - you suggest I don’t understand what I’m talking about but you can’t give even the tiniest hint as to where such an alleged misunderstanding might originate.
Do you think that all is needed is to point out errors? Is it not possible that when you tried to get your ideas published, the people who you sent them to believed with some reason that it would be futile to point out errors because when they tried this in the past with other people who they considered cranks, those people just could not or would not understand what was being pointed out to them? And that they suffered from some sort of strange irrational blind spot that made this difficult for them and made trying to deal with them a waste of time?
That is one possibility. The other possibility is that every single course taught in every math department everywhere, in teaching Gödel’s theorem and contemporary proofs of it, is wrong, and you have discovered something that none of them can see. Does that honestly seem likely to you?
James R Meyer
I agree that the phrase “absolutely everyone” is too extreme, and I didn’t intend to make that claim, that is why they word “seems” is there. However, let us try to make a precise claim. I suspect that, quite probably indeed, absolutely every mathematical logic course in every accredited US university that deals with Gödel’s theorem accepts it as true (in a contemporary form) and virtually all such courses prove it as well. Can you name any counterexamples to this? In fact, I would love to see one! There will be some people who do technical logic at a high level which do not accept it, in the sense that they do not even want to accept the logics that it is talking about (because they are finitists, or intuitionists, or something like that) but that is not what we are talking about here.
As for your complaint about people complaining about your website, I tried at length a few years ago to give you hints, and eventually you just started to ignore me. So I offer again, let us go through it all rigorously and carefully and find exactly where there is a problem. I suggest we start with the so-called “Proposition V” since that is something you have some problems with. Proposition V is a pretty trivial piece of mathematics, although very tedious. If we limit ourselves that that, and find no problem with it, would you agree to reevaluate your entire position? Wouldn’t you like to know the truth?
If I recall you have issues that go even deeper than any specific problems with Proposition V. Your issues may go down to the very nature of mathematical reasoning itself. I think your objections to Proposition V will end up being objections to something even more fundamental and totally unrelated to Gödel.
James R Meyer
In your discussion of Peter Smith’s proof you say that the function diag that he gives is not primitive recursive. But it is.
The reason you give for your claim is that Smith says diag(n) is the Gödel number of the sentence obtained by taking the sentence whose Gödel number is n and substituting into its one free variable the numeral for n, i.e. the term formed by n S’s followed by one 0 (and is just n if n is not the Gödel number of a sentence with one free variable). You seem to be saying that since this characterization refers to non-numbers the function cannot be primitive recursive.
But that is simply a mistake. A function is primitive recursive iff it maps from numbers to numbers and *can* be expressed in a certain form involving only numbers and other primitive recursive functions. But that does not at all mean that the function cannot *also* be expressed in *other* ways that refer to things other than numbers, or use methods other than primitive recursion. In fact this is a crucial fact in the proof of incompleteness theorems.
A trivial example could be the function f(x) = “the number of times the letter ‘e’ appears in the word ‘eventually’ ”. Clearly this identifies a primitive recursive function, the function f(x) = 2.
James R Meyer
Here Nova Kant demonstrates a common fallacy, the origin of which is by making an unfounded assumption rather than following a logical chain of implication.
It is correct that one can have a p.r. function that is f(x), where f(x) is a term that is a name for an expression such as:
2 + (0 times x)
where x is a variable whose domain is natural numbers and the function evaluates as 2 for x substituted by any natural number. Such a function is easily expressed by a formal arithmetical system.
And it is correct that one can have a function that is given by the expression:
the number of times the letter ‘e’ appears in the word ‘eventually’
and which gives the value of 2,
and one can give it the same name of f(x). (see NOTE A below)
But simply giving it the same name - which is all that Nova Kant is doing - does not somehow magically confer upon it the property of being a number-theoretic primitive recursive number function (see more below).
I have already explained in my paper on Smith’s proof (the paper that Nova Kant is referring to (Footnote: See A Fundamental Flaw in an Incompleteness Proof by Peter Smith for the paper on An Introduction to Gödel’s Theorems by Peter Smith, Cambridge University Press, 2006, ISBN: 9780521857840, An Introduction to Gödel’s Theorems: details here. )) that equivalence of evaluations of two different functions does not necessarily mean that other properties of those two functions are necessarily also equivalent. (see NOTE B below)
Note that Nova Kant seems to think that somehow that the expression:
the number of times the letter ‘e’ appears in the word ‘eventually’
can be expressed by a formal arithmetical system, by an expression such as:
2 + (0 times x).
But it should be quite obvious that the expression:
2 + (0 times x)
does not capture the compete information contained in:
the number of times the letter ‘e’ appears in the word ‘eventually’
Of course, Nova Kant has tried to facilitate his argument by choosing an expression that omits any variable that has a domain that is not natural numbers, (note that for incompleteness proofs such as Peter Smith’s, such a variable is required) but we can correct that by a function such as:
f(ω) = the number of times the letter ‘e’ appears in the word ω
where the variable ω has the domain of words in English. Now, while the function gives a numerical quantity upon a correct substitution of the variable ω, there is no function in a formal arithmetical system that can express the information content of f(ω).
Note A: Incidentally, Nova Kant is technically wrong in using the name f(x), which includes the free variable x, for the expression “the number of times the letter ‘e’ appears in the word ‘eventually’ ” as f(x), since x does not occur in the expression that the name f(x) is being applied to.
Note B: Page 12, Section 6 of the paper A Fundamental Flaw in an Incompleteness Proof by Peter Smith:
“But any assertion of equivalence/equality is an assertion that the properties of the entities for which equivalence/equality is claimed are identical within the context of that assertion. So while it may be correct that an assertion of equality of 6.1 and 6.2 is correct with regard to the property of numerical value in the context of a system comprising of the rules and axioms of arithmetic together with the definition of the Gödel numbering system, that assertion of equality does not apply to the property of being a primitive recursive number function.”
- which is the illogical and unfounded assumption that Nova Kant is making. He provides no logical argument that the assertion of equality of evaluation implies that other properties, in particular the property of being primitive recursive, are also necessarily equivalent. That is an elementary logical error where the desired result is made by an unfounded assumption rather than given by a correct logical implication from one statement to another.
James R Meyer
Note: Nova Kant’s posts are becoming too large for this home page. Any further comments on this thread will now be at the page A typical discussion about my work.
You make it clear that the way mathematicians write can be confusing for people who are not themselves mathematicians. I will try to clear a few things up here.
The expression “f(x)” is a compound expression that refers to the value of function f when applied to x. It is not itself a function, p.r. or otherwise or even a name for a function. The function here is named by “f”.
When people write
f(x) = x + 1
to tell others what they mean to refer to by “f” this is not the best notation and it can confuse people, because after all, what is“ x”referring to? They really mean to say
(x) f(x) = x + 1
which, presuming we are talking about functions from N to N, is a statement about f which satisfied by only one function and consequently could be considered a definition. This however is still not the best way to do it. It would be better to give a term that denotes the function and simple say that our function of interest is equal to that term. So the best thing to say is
f = lambda x. x+1
or, as Harrison’s HOL-Light and other systems notate it:
f = \x.x+1
Please notice that it is not at all necessary in any of these cases to use the formal parameter “x” in the expression defining the function. So we simply have what is called a “constant function” which could be defined by saying
f = \x.2
This is not, as you say “technically incorrect” as it is not at necessary to use a strange locution such as
f = \x.(2 + 0*x)
To continue, the functions f and g defined by
f = \x.2 and
g = \x.(the number of ‘e’s in the word ‘eventually’)
are the same function. In other words,
(the number of ‘e’s in the word ‘eventually’) = 2
The condition for equality of functions is simply
f=g iff (x) f(x) = g(x)
This is the standard that is used throughout all of mathematics which people are taught in elementary school. So the disagreement you have with Gödel’s theorem is a disagreement about some basic elements of standard mathematical practice. Your objections arise even before we get anywhere near anything unusual in Gödel’s theorem.
The point of a function is simply to associate each element in one set (the domain) with a unique element in another set (the range). You use some of these terms so you seem to agree with that much. Even if one does not identify a function with a set of ordered pairs, and inherit the definition of equality from the definition of set equality, the condition for equality given above is still the one that is used.
Now getting back to primitive recursiveness a function is going to be primitive recursive only if the domain and range are the natural numbers (although we can certainly extend the concept involved in primitive recursion to other sets but at this point perhaps that would just add additional confusion).
This means when someone writes (which would be silly frankly but I am using that example to make a point):
f = \x.(the number of ‘e’s in the word ‘eventually’)
they are simply concerned to specify f as a function from N to N. They are not concerned to “capture all the information”, as you say, in the definition of the function.
I did not need to use a constant function either, in order to make this point. But of course I could not have used your example:
\w.(the number of ‘e’s in the word w)
since this is not a primitive recursive function because it maps words to N, not N to N.
So now back to diag, which *is* a function from N to N. And it is primitive recursive, it can be proven to be so. However, I could also write out a specification like this
diag = \X.
take X and interpret it as representing a sentence S in the language of arithmetic according to the following conventions:
check to see that this is a sentence that has only one free variable
return the number X
form the sentence S1 that results from taking the term “SSSSS… X times…. SSS0” and substituting it in for each occurrence of the free variable
now turn S1 into a number X1 using the same conventions as above (applied in reverse)
return the number X1
From this definition of diag, we can see quite clearly that diag is a function from N to N. However, the specification uses internally various concepts about formulas and such. I repeat that there is absolute nothing wrong with doing this when defining a function, if one has access to these concepts. However, what it means is that it is not immediately evident that diag is primitive recursive. To show that it is, we would have to first write out the specification more clearly, and then show that we could write out a specification for another function, call it g, that was in the correct form, and then prove that f = g using the definition of equality for functions
f=g iff (x) f(x) = g(x).
And this is, in fact, what is done, although not completely formally, because that is quite onerous and tedious although not mathematically very difficult. This has to be done when proving Gödel’s theorem because we need an expression that can be written in the language of arithmetic which specifies the function diag, and of course we do not have direct access to the concepts of sentences and formulas in the language of arithmetic. (There are two issues here. We have to first express the function using the language of arithmetic, but we must also show that the function can be *represented* in the axiom system as well).
If one wants something like this in the case of Gödel’s theorem, both Harrison and O’Connor have written completely formal proofs in the systems HOL-Light and Coq. We should be grateful to them and the other professional mathematical logicians who have labored long and hard to add to our collective human understanding of reality. I have not looked at O’Connor’s proof but I looked at Harrison’s and I don’t like it much because it is very difficult to read, because of the way he employs backward tactics in the proofs and the intermediate formulas are not written out. These tactics only expand into the formal proof when it is executed, and that proof is not shown. I actually wrote code to alter the core of Harrison’s system to save the complete proof as it is being generated, and it is not easy to read but it would be possible to read it and check each step.
Perhaps we should write up our own formal version that is easier to follow, pedagogically speaking? How about it?
James R Meyer
The first part of your post is irrelevant verbose waffle about conventions re formatting of functions.
The principal issue is how a formal system, that has only variables that have the domain of natural numbers, and sets of natural numbers, can have a representation of a statement that is in a different language. This is commonly called representability.
If a statement in that different language includes variables that do not have the domain of natural numbers, or sets of natural numbers, and where such variables cannot be eliminated from that statement, then that statement cannot be expressed in the formal system.
Also note that all symbol sequences that belong to the domain of the variables of the formal system must be natural numbers in that language, or sets of natural numbers in that language. That requirement is not commonly stated in the definition of primitive recursive but it is implicit that a valid expression of a language system must only consist of valid symbol sequences of that system.
Now, your original claim was that Smith’s function diag is primitive recursive, and hence representable in the formal system. You claim that if a function maps from natural numbers to natural numbers, then it is primitive recursive, and hence representable.
But you then waffle on about Smith’s diag, and say things like:
“…it is not immediately evident that diag is primitive recursive. To show that it is, we would have to first write out the specification more clearly…”
But Smith has completely defined diag in his paper. You say that I am wrong to say that Smith’s definition of diag is not primitive recursive, but all you have to offer by way of argument is by waffling on about anything other than Smith’s definitions.
And, as I point out in my paper A Fundamental Flaw in an Incompleteness Proof by Peter Smith, in order for Smith to claim that diag is primitive recursive, he has to use the “equality”
num[n] = GN[overbar(n)]
But the variable of GN has the domain of symbol sequences of the formal system.
I see you have retracted without comment your claim that I never made a reply when in fact I replied promptly nearly a month ago.
“The first part of your post is irrelevant verbose waffle about conventions re formatting of functions.”
Sorry, it is just my sincere attempt to clarify some issues that seem to me to be confusing you. At the very least, hopefully you will see that there is no problem defining a function using a terms such as \x.2 rather than the unnecessarily awkward \x.(2+0*x).
“The principal issue is how a formal system, that has only variables that have the domain of natural numbers, and sets of natural numbers, can have a representation of a statement that is in a different language. This is commonly called representability.”
This may be how you use the term “representability” but it is not how it is normally used in mathematical logic. First of all, it is hardly a definition since it refers to the undefined term “representation” itself, and uses a vague verb of possession “have”. But more importantly, representability (or as our author calls his version, “capturability”) is not at all about statements in “different languages”. All the statements involved are in the same language, which here is the language of arithmetic.
For example, consult the SEP article on Gödel’s theorem where various strengths of representability are defined:
“A set S of natural numbers is strongly representable in F if there is a formula A(x) of the language of F with one free variable x such that for every natural number n:
n ∈ S ⇒ F ⊢ A(n);
n ∉ S ⇒ F ⊢ ¬A(n) ”
Please check the article, the formatting doesn’t come through perfectly (the underbars are missing. The “n”s in brackets are nothing more than the standard numerals for their corresponding numbers -- as close to the same language as you can get). The point is that there is no other language here, it is all just the language of arithmetic and we are just talking about numbers and sets of numbers.
To give a specific example, we might ask if the property of primality (or equivalently in this context the set of primes) is representable in a given theory F.
First we should observe that according to the standard semantics for the language of arithmetic, primarily is *expressible* in the language with the following formula:
x>1 and “(y1)(y2) y1<x and y2<x” and “y1*y2=x” and “not y1=1 or y2=1”
(Strictly speaking this is not in the standard language of arithmetic because it contains the symbols 1 and < and it takes some informal liberties with syntax but I will ignore that for now. All of this is covered very well in Peter Smith’s excellent textbooks)
Second, we should note that this is not enough for representability. For representability we want to be able to *prove* the true sentences of this form and prove the negation of the false sentences of this form. So, if this sentence is called PRIME, and we write in short form PRIME(x) for this sentence with the numeral for x substituted in for the free variable, in order to have primality represented by this sentence in F we need to be able to prove from our theory F the sentence PRIME(sss0) and the sentence“ not PRIME(ssss0)” as two examples. Of course for primality to be representable in F, F would have to prove an infinite number of statements of this form as should be clear from the example above.
All of this is hopefully to help you understand what representability is, compared to what you said above. First it is not about“ having representations” (whatever that means) and second is not about one language saying something that was expressed in another language.
“Now, your original claim was that Smith’s function diag is primitive recursive, and hence representable in the formal system.”
Yes, okay, but the “hence” is not an obvious thing. One has to prove that all primitive recursive relations are representable. It is not obvious and in fact the proof is a little tricky. However, we should also note that we cannot just say “the formal system”. For example, it is much easier to show that primitive recursive functions are representable in PA, but Smith and others take on the more difficult task of showing they are even representable in the far weaker system Q of Robinson Arithmetic.
“You claim that if a function maps from natural numbers to natural numbers, then it is primitive recursive.”
No, I don’t think I claimed that and certainly did not mean to. A function mapping from natural numbers to natural numbers is a necessary condition for it to be primitive recursive, not a sufficient one. (Although as I also pointed out, the restriction of primitive recursive functions to natural numbers is just a relatively arbitrary convenience that developed in computability theory.)
“You say that I am wrong to say that Smith’s definition of diag is not primitive recursive”
No you are right to say that the direct definition Smith gives of diag is not a primitive recursive definition. He just gives a definition that shows why we need diag, what it “does”, in effect. But this definition is not in primitive recursive form. However, that does not mean the *function* is not primitive recursive. It is. For a function to be primitive recursive, as I already pointed out, is for it to be *possible* to give a definition of it which is in primitive recursive form. That is something he does not do explicitly, but suggests quite clearly how it could be done. You are confusing whether a function is primitive recursive with whether or not a definition is primitive recursive. Those are two different (but related) things which you have to distinguish.
To perhaps clarify more, you can consider that there is another kind of function definition talked about in this context, that of sigma-1 definability. A function can be called sigma-1 defineable if it is expressible by a sigma-1 formula. All of these things have to be kept carefully separate in order to prove anything rigourously in this context.
Again all of this is covered in Peter Smith’s excellent book. I would suggest if you want to learn the truth about Gödel’s theorem that we take this book and work through it systematically clarifying it as we go along. I made the suggestion earlier but it was ignored. However, since I do have some other things to do, perhaps the effort involved in such a project is not worthwhile. Unless maybe there are some others reading who also want to contribute? I haven’t seen any evidence of that.
James R Meyer
Re: “I see you have retracted without comment your claim that I never made a reply when in fact I replied promptly nearly a month ago.”
Yes, because it was lost in my Disqus spam, and it was only by your drawing my attention to the fact that you had made a post that I discovered it. I was very impressed by the fact that Disqus was able to recognize complete hogwash.
You assert that my reference to representability is vague. Yes, of course it is, and the reason is this. In incompleteness proofs the notion of “representativity” is often used - instead of providing a completely logical step by step proof and actually translating a given statement into a statement of the formal system by using a translation function, and thereby proving that such a translation is possible. And as I point out elsewhere on this site, there are claims of “completely computerized proofs” which do not actually provide a complete logical step by step proof, and instead rely on a notion of representability.
So don’t try to offload the problems associated with a definition of representability onto me. I didn’t invent the notion. It is used by people who are trying to prove incompleteness proofs, so it’s up to them to give a rigorous definition, not up to me. You can generate a proof that can satisfy any definition of representability that you like to make, and you can prove the “representability of statement A” according to your definition of representability, but that of itself does not prove that there is an expression in the formal system that states exactly the same thing.
Therefore you also need to prove that your representability is exactly what you claim it to be - and so the onus is on the person who is claiming an incompleteness proof to provide a totally clear definition of representability - not me.
And don’t try to deflect the issue here by referring to Stanford Encyclopedia or whatever. If all you are interested in is appeals to authority rather than logical argument, then go elsewhere. The ethos of this site is for logical argument with due consideration of how incautious use of language can result in errors of logic - and not for simple-minded lapping up of the unproven claims of people such as Panu Raatikainen (who wrote that SEP article you refer to) who thinks that his article https://plato.stanford.edu/entries/goedel-incompleteness/sup2.html is a proof, but where he simply leaps over any difficulty by simply asserting “The latter operation can be expressed in the language of arithmetic” rather than actually proving it (see also Panu Raatikainen’s Proof of the Diagonal Lemma).
Considering that you criticize my reference to representability as vague, it is risible that you yourself have the audacity to say:
“All the statements involved are in the same language, which here is the language of arithmetic.”
That’s ridiculous. The “language of arithmetic” is a vague term that does not refer to any one specific language system - it doesn’t have a specific alphabet, nor a specific set of rules of grammar, nor a set of rules of inference. Hence it might be more correct to say “the languages of arithmetic”.
As for your idea that all statements involved are in the same language, the fact is that incompleteness proofs necessarily have a delineation of language of meta-language and language, as indicated by the use of a Gödel numbering system, which necessarily uses a free variable that has the domain of symbol sequences of a sub-language.
As usual you decline to go into any detail regarding Smith’s actual book and my actual paper about that book.
But I will. First see Smith’s Theorem 19.2 in Section 19.6 (2nd Ed of the book) where he defines a function called diag(n) as:
(1) Exists y(y = numeral (GN(Phi) AND Phi)
where Phi is a variable whose domain is symbol sequences of the formal system.
This function must always result in a natural number for a substitution of its free variable, since GN always results in a natural number, and hence the range of the above function is a set of natural numbers. The domain of the variable of GN is the set of symbol sequences of the formal system. Hence the domain of the function is symbol sequences of the formal system, and its range is natural numbers.
You suggest re Smith’s function diag that it is:
“possible to give a definition of it which is in primitive recursive form. That is something he does not do explicitly, but suggests quite clearly how it could be done”
You are wrong. Smith did give an explicit definition of a function called diag(n) that is primitive recursive. In his Theorem 20.2 in his Section 20.1 he claims that a function called diag(n) which is:
(2) 21^3 · 3^4 · 5^17 · 7^4 · 11^15 * num(n) * 2^3 * n * 2^19
is the very same function as the function diag(n) he previously defined.
Here the domain of this function is the set of natural numbers, and its range is a set of natural numbers. Hence the function is a mapping of:
- the set of natural numbers to
- a set of natural numbers.
But according to you and to Smith, the very same function is also a mapping of:
- a set of symbol sequences of the formal system (and which are not all natural numbers) to
- a set of natural numbers.
Perhaps one might try to get around this by saying that (2) is a partial function of (1). But since it is only (2) that satisfies Smith’s notion of being representable in the formal system, any proof of that representability must not depend on any property that is only of (1). Hence, if the proof is correct, it must be possible to eliminate any mention whatsoever of (1). Good luck with that.
Okay, I have spent many stressful hours calmly and patiently and with good will trying to help you resolve your difficulties with Gödel’s theorem, and you constantly come back with replies full of insults such as claiming that Disqus was able to detect that my post was “complete hogwash”.
I really was very interested in trying to help you gain an understanding and appreciation for a crucial and central result in mathematical logic, and in doing so deepening my own understanding of these issues, but the constant barrage of disparagement, insult, and lack of appreciation for sincere effort which I am required to face has forced me to reconsider.
James R Meyer
My insults? You conveniently forget that I tolerated your insults in your initial posts, when I could have blocked you.
Your first post told me I do not understand what I am talking about, and your next post implied that I am a crank and not able to understand what is pointed out to me. Your next post said that I “have issues that go even deeper than any specific problems with Proposition V. Your issues may go down to the very nature of mathematical reasoning itself.”
Two posts ago, you tried to sidestep my discussion of Smith’s paper by replacing Smith’s definitions by your own. And instead of pointing out any faulty step in my step-by-step analysis of Smith’s paper, you talked in general terms about it - but you couldn’t even get that right - in your previous post, you claimed that Smith did not give an explicit definition of diag as primitive recursive.
In fact he did.
So I don’t apologize for describing what you wrote as hogwash - because it is.
Is it any wonder that by that stage I was fed up with your posts? If you post comment after comment in which you avoid engaging in logical argument on the actual facts and are wrong in basic details, what do you expect?
You’ve chosen a convenient excuse for dropping out when you have discovered you simply don’t have the capacity to engage in detailed logical argument, but I’m not fooled by your chicanery.
It’s rather pathetic that this correspondent asserted that he was trying to help me “gain an understanding” by referring to Smith’s book when it is clear that he has not made himself sufficiently cognizant with that book, and when he believes that one can have two valid definitions of a function, where one definition defines the function as having a particular property and the other definition defines the function as not having that property.
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