Math And Objective Existence

{Can you give a "clear enough" definition of what a function returns?}

I don't want to define stuff if I don't have to. Based on past experience, I should try to avoid attempting rigorous IT definitions. It seems we can only say concrete things about explicit implementations, not general concepts. But I will say that in D1 languages (dynamic with explicit types), functions return at least a type indicator of some kind, and a value/literal/data-item-like thingy. The packaging of these two concepts differs per model, and as long as these two things are "kept" for a while in some manner and are accessible to the model's guts, the models have sufficient info to "do their job". --top

[No. You just refuse to learn how to say concrete things about general concepts. The rest of us have been doing so for thousands of years.]

You just think they are concrete because you mistake your personal head model as universal/external truths. You have a mental block problem in that area and don't recognize subjectivity. Your alleged concrete statements just define vague terms in OTHER vague terms, creating FractalVagueness.

{So you say.}

[Primes are a general concept. What's subjective about them? Is there something that's a prime for one person but not another? Or how about something that's a prime for me today, but might not be tomorrow?]

Note that I never claimed primes "change" from day to day. Claiming them to be "subjective" does not mean I have to accept all mass chaos scenarios that can potentially be associated with them. Perhaps one can say that subjectivity is a continuum.
[But you are claiming they change based on the person considering them. That's what subjective means after all.]
I'm not sure I subscribe to your definition of "subjective", but yes I suppose somebody could wake up one day and reject the existence of primes because integers are not a real thing, per below. They may invent or prefer to use "fuzzy primes" based on approximations of integers and/or probability of accuracy, not unlike FuzzyLogic compared to BooleanLogic.
- Note that I interpreted "prime to one person but not another" to be a claim that's a test of subjectivity.
- [Noted. And what you've been asked to do is to come up with such a prime.]
- Where, I missed it.
- [Just go to the root paragraph this is attached to.]
- I already answered that above. You want an example of a specific "probabilistic prime" or a "fuzzy prime"? I don't have one ready. It's not an unfathomable situation, though. And as I mentioned elsewhere, somebody may reject the existence of primes altogether because integers don't objectively exist. So, you want me to come up with a non-prime for that situation? Okay, Here it is: ___
- But I personally am willing to except the existence of integers as a working assumption such that the above is not a comparable analogy to our "semantics" debate. You are trying to trick me into a political manipulation trap whereby you say, "Top rejects integers, so how can he say anything important about languages?".
- [No, you dodged the question by trying to change the subject to lies and reality. (BTW, I didn't ask for a non-prime, I asked for something that is a prime for one person and not another even when everyone is using the exact same rules, notations, and definitions. That's what it would take to support your claim.]
- I'm not lying, you bastard befuddled outdated detached Alzheimered professor! Your "same rules..." criteria is artificial. It relates to no claim I've given.
- [I didn't claim you were lying here. I claimed you tried to change the subject to lies. There is nothing artificial about the "same rules" criteria. If they were using different rules, they would be talking about different things. The claim you made requires that they get different results sometimes even when they are talking about the same thing. Hence the "same rules" criteria.]
- Hmmm, so you meant the denier of objective integers is a "liar"? If so, I apologize for interpreting it as a personal accusation. And I wouldn't call such a person a liar. I too think integers are merely a UsefulLie and don't objectively exist in reality. For example, no two bricks are exactly the same size such that some are "more brick" than others. However, we kind of round them them to the nearest integer (based domain-specific spans) for our accounting UsefulLie's.
- [No, I don't mean that. You started talking about lies when I asked you the questions you've been dodging. I did call you a liar elsewhere when you denied saying things that you had, in fact, said. But that wasn't in what you responded to here. And once again, the existence of a physical structure that matches the abstract structure of the integers is immaterial.]
- I'm confused. Need Excedrin. And yes, it is "in the head".
- [I agree, you are obviously confused. And no, it's not in the head, it couldn't possibly fit since it's infinite and heads are finite.]
[That would be somebody talking about something else while using words that already have a different meaning. That does not show that they are subjective.]
You realized you used "have" again.
[So? It correctly conveys the meaning I wish it to convey.]
So you think. There is no known God of Definitions that directly inserts meanings into words like a vet giving a cat a shot. The meanings are conveyed by human minds processing words; the word's don't dip themselves into the central repository of meaning vats nor are dipped by God. When you say "words have meaning", this have-ness happens in individual heads. There are different ways to mentally and paper-wise (?) compute primes, which suggests the notions in different heads about what primes "are" may not be universal. I'm not claiming there is absolutely "no primes"; I'm only claiming their objective existence is probably impossible to verify outside of the perspective (scope) of a mutually-agreed-upon notation system. (Semantic equivalence has similar problems.)
[So I know. In natural languages, words are defined by how the population in question uses them. These definitions (after an understandable delay) are reported in things we call dictionaries. In a sense, we, collectively, are the "God of Definitions" you rant about. But, in the end, this is just another attempt to dodge the questions. In addition, no one has said that you were claiming there are absolutely no primes. That is also a dodge. And for your final point, the existence, as an abstraction, of primes, is verified by noticing that primes have been used for thousands of years.]
Dictionary writers study usage and do their best to produce a spoken-language definition or description, yes, but this is a subjective process. And I don't know if primes are subjective or not, we just haven't found a way to objectively verify their existence beyond something tied to specific notation. We can't measure prime-ness in a universal sense, but only in the "world" of a specific notation. Not being able to verify objectivity is not necessary subjectivity. We just need to be honest about what we really know and can measure versus ideas that may "feel" universal (super-notational). Thus, I cannot answer your question as stated. I'm only claiming their objectivity has not been verified, not that primes are subjective. -t
[The process is, by and large, not subjective. And yes you did claim that primes are subjective. You said, in bold, "Primes are subjective." Now quit dodging and answer the questions.]
If I said that, I take it back for now (putting the existence of integer issue aside for now also). I don't know their status. My official stance is that they haven't been proven objective. They are defined in terms of given notations, and nobody has shown that a definition in one notation is equivalent to all others in the same or different notations that have "integers". Nor can I practically tell if each person has the "correct" definition in their head. Thus, I cannot answer questions about what's in other people's heads other than estimates.
[You most certainly said it. Not only that, but you said it on this page recently. I don't think anyone really cares what your official stance is since you've shown a shocking degree of ignorance over such subjects. (You've quite obligingly provided evidence of this with your rant about notation.)]
Projection. You have no formal proofs, only ArgumentFromAuthority.
[Thank you for providing more evidence of your ignorance. I've not used a single ArgumentFromAuthority on this page, neither have I presented a formal proof. What I've done is asked you to prove, informally, one of your claims by actually coming up with the thing that must exist if your claim is true. Now quit dodging and answer the questions.]
False dichotomy. Not being able to prove them subjective is NOT the same as proving them objective. The actual label is something like "unknown" or "null".
[Since there's no dichotomy above, there can't be a false one.]

Some argue that integers are a UsefulLie approximation and don't reflect reality. We can count "people", but people are full of many different life-forms such that a biological definition won't settle anything. And cancer and mutations rule out using DNA alone as the cut-off between human and non-human or "belonging" to the same individual versus not belonging. The bottom line is that two parties will agree on one or more UsefulLie and then build to the next step based on that set of agreement. If my acceptance or rejection of UsefulLie's does not match that of the reader, then they will probably disagree with my conclusions. But I don't think my stated assumptions are abnormal or unreasonable to most readers, or at least "regular programmers", who the target model is designed for.

[Whether or not integers reflect reality is immaterial. Please answer the questions instead of dodging them.]

Primes depend on the existence (assumption) of "integers" such that it is material.
[No, the existence of a physical structure that matches the abstract structure of the integers is indeed immaterial. Now quit dodging the questions.]
I assure you I am not intentionally dodging anything. If the physical existence of primes depends on the physical existence of integers, then they are indeed tied. The "abstract" existence is a head notion, and usually requires an agreed-upon notation (maths) to serve as a representation to be able to form and verify any concrete statements about such. We can do rigor on notations, but it's very difficult to do rigor on thoughts alone. We can assume there is a more centralized, abstract, or universal notion/idea/concept behind primes than specific notations, but such is not measurable outside of notation. Such beyond-notational (super-notational?) is a hypothetical "substance" that cannot be measured. We can't do "math" on notions alone (for others to study); we have to convert them into notation first.
[That would be easier to believe if you varied you dodges. Once again, the physical existence of integers is immaterial. The physical existence of primes is immaterial. The fact that we need a language to communicate with each other is also immaterial. Please quit dodging and answer the questions.]
Exactly what questions am I allegedly dodging? Give them PageAnchors with unique names/ID's please. Explicitness is not your strong point.
[Look for "Primes are a general concept."]

Primes ARE subjective: they don't exist in reality. They are a head model. We have various notations for math that result in physical markings, but these are just a projection of mental ideas; a mutually-agreed-upon representation of them with mutually-agreed-upon rules. Math is not science. If there is a dispute over claims about primes, two or more parties use the agreed-upon representation/notation and related rules to work out the differences. We even have machines (software) that can successfully (agreeably) process math because almost everybody agrees with the notation and related transformation rules.

Contrast this with politics. While both sides ("left" and "right" here) agree that "we should improve the economy", both sides don't agree on the process/transformation to do that.

[Whether or not integers reflect reality is immaterial. Please answer the questions instead of dodging them.]

I'm sorry, I thought I did, with explanation.

[No, you just repeated your earlier rant using different words.]

I'm sorry, it looks answered to me. I don't know what else to add. Try different or more specific questions or specific scenarios/examples/cases perhaps.

[Try answers to the questions I asked. Subjective doesn't mean "does not reflect reality". Subjective doesn't mean "not real". (And how can I get more specific than asking for a number that is prime for one person and not another?)]

Are you asking me to estimate the likely responses to a survey about primes given to multiple people?

[No. I'm asking you to show me something that meats the definition of prime when considered by one person, but doesn't when considered by another. They must be using the usual definitions and rules for such things yet come up with different results. That's what it would take for primes to be subjective.]

"Using the usual"? Isn't that ArgumentFromPopularity? (Also see above re "fuzzy primes".) As far as agreeing to the reference notation and agreeing to certain transformations, but not further results based on those transformations is fairly rare and not the kind of problem that we usually encounter here. Our usual problem here is agreeing that some specific notation or "root" rules about such notations reflect "common notions" as originally given in English. Our problems are usually translating English into agreeable base notations and/or transformation rules, NOT subsequent steps based on that base set. Even something as seemingly simple as "values have types" get into a big disagreement on how to represent/process "have" and when this have-ness is active or not in a model that attempts to mirror human notions.

[No, it's not an ArgumentFromPopularity. It's not even an argument. Please quit dodging and answer the questions.]

PageAnchor ET-Apple

I'm honestly not dodging your questions. I believe the problem is, based on years of trying to read your material, is that you are a poor writer and don't realize it. Part of this is that what you write in English is a lossy version of your head model, but you don't realize this is the case because when YOU read it back to yourself to check it, you unconsciously re-inject into it the "lost" pieces by filling in the gaps with your existing head models. See TheMartianAndTheApple for an analogy. I believe you to be unconsciously making a similar mistake. My interpretation of your English statements does not match yours, despite not being "wrong".

[Actually, if your interpretation doesn't match mine, you are indeed wrong. That doesn't mean it's your fault, though in this case it would be since the definitions involved have been used by almost everyone for thousands of years. (I'm talking about "prime" and "integer" here, not "subjective". I don't know how long that's been in use.)]
But that "thousands of years" is perhaps equivalent to the experience with the apple. And the focus should be on communication improvement, not on "who is wrong". The apple fax is a lossy representation of an apple, and your English descriptions are a lossy representation of your thought process. Perhaps others with a similar background will interpret them the same way, just like Earthlings will think "apple" of the fax; but that does not mean the writing is good, but rather that it's tuned fairly well for an audience with a given experience set (probably similar to that of the author). The fax artist could be an asshat and say, "I don't care if aliens interpret it as something else; the drawing itself is not objectively wrong! My peers know what the hell it is, and that's good enough; screw everybody else who doesn't walk in my shoes!" The fact remains that the drawing is ambiguous even though your peers are not thrown off by the ambiguity. This can generate a false sense of clarity or universality. The real fix is to find a less lossy or less ambiguous way to represent/convey apples, not argue that your first attempt is "good enough" because it works for your buddies.
[My focus has been on getting you to stop dodging the questions I asked.]
Then ask better questions, dumbass.
[What's wrong with the questions?]
Give them labels or numbers for one, so I know what you are referring to instead of have to rely on pronouns.
[That has absolutely nothing to do with how good the questions are. I've also told explicitly how to find them, so there's no problem with how I'm referring to them. Now quit dodging and answer the questions.]
Your instructions suck. I answered all the ones I've found.
[I can't help it if you can't perform a text search.]

{I don't think that's it. You don't appear to have made any attempt to answer what are very straightforward questions. This has all the appearance of evasion. If you don't know what prime numbers are, just say so.}

You really think they are "straightforward"? "What they are" is simple English, but "is" and derivatives are not "straightforward". Classification or equivalency is quite often a source of contention. It's only "clear" (rarely disputed) where an agreed-upon notation and related base rules exists. There are probably multiple formal definitions of "primes". However, that does not necessarily mean a given person "runs" that formal definition in their head. For example, a security technician who uses prime numbers for their work may not care about how they are generated, but views them as something akin to "list of funny numbers I get from an online table that I use to manage security tools". To him/her, it's merely a "source" of certain kind of numbers useful for a given purpose. If you probe them, they may be able to give you a typical-sounding definition of having no non-one divisors, but his PRIMARY mental model does not revolve around their "generation" criteria.

Reminds me of this story:

National Geographic Writer: "Okay you gays, can you describe to me what a bison is?"

Human: "A large, powerful, fury grazing mammal."

Elephant: "A small furry animal that keeps muddying up my favorite watering spots."

Mouse: "That big brown shadow that squashed my brother!"

Lion: "The best food!"

To most humans, primes would probably be described as something like, "A number that satisfies that annoying paper process [algorithm] I learned in school to pass the test".

{You appear to be conflating personal feelings about mathematics with mathematics. How odd.}

Are you asking me for a formal mathematical definition of "prime numbers"? Otherwise, "is" is relative; Classification (set membership) and equivalence is based on perspective.

[Of course not. We are asking for you to answer the questions instead of dodging them.]

Well I do agree with that. The few times I have thrown anything into the discussion, what I have thought I was saying somehow changes when he replies. It is as if he is trying to avoid being clear in his answers. I really dunno. I look in regularly to see if there is any progress, and there is none. Just more pedantic (IMHO) argument. I still don't know what a tag offers, and where it may be used. --AnonymousDonor

Well, I tried my best to answer the questions and explain the design tradeoff choices as carefully as I can. I don't know where the communication breakdown is happening. Your criticisms are usually at the summary level, and not the detail level (choice weighing considerations) such that I cannot apply them to anything concrete. My variable/value is less characters than the other "nested" equivalent; that has to at least give it some objective points. And "is" is relative. There is no known central classification force in the universe. A bison "is" food to a lion's primary classification system. And most rigorous proofs come after a notation and its base rules are vetted and agreed upon. Until such is formed for "semantics", you are in foggy territory where few if any other fields have done well in as far as good science. Your claims and questions ofen assume canonical/central objective standards that have not been clearly established or measured.

One may have multiple definitions in multiple notations for "primes" that can produce or qualify numbers as expected in our tests; and intuitively most feel there is a "central" concept to them even though the notations and definitions vary. However, this "central concept" has not been scientifically tamed. I'm just the messenger.

Well there ya go! Long-winded multi-topic confusing irrelevant, if the above is meant to be any response to the AnonymousDonor. Whatever you are trying to say, you are not saying it. Years ago the AI people launched neural networks which they intended to train to participate as humans on the web. I am starting to think that Top may be one of them. i am convinced that who or what ever he is, he is not honestly trying to communicate an idea. Or is it that he is insecure about letting it out where he has already been strongly refuted on just about everything he has said - from functional programming criticism to tag promotion and type demotion. Once again I have decided to keep out of it. Good luck. --AnonymousDonor
There's no "strong refutation". You would have had formal logic with numbered clauses kicking my ass by now if you had such. Your reasoning is often based on (false) WetWare assumptions that don't have the clinical evidence behind it. (Either do my WetWare claims, but I admit it.) It's just the expensive universities trying to justify expensive education by trying to make stuff unnecessarily arcane and obscure. Your self-worth and/or wallet is threatened when somebody claims that objects are good enough such that HOF's are not necessary (ParadigmPotpourriMeansDiminishingReturns), or propose an alternative type model that is more approachable because it uses imperative code and XML instead of the Chris Date approach. It's the Catholic Church in the face of small challengers to the (alleged) canonical world view. I don't expect the current generation to realize their greed and stubbornness, and you fuckers will continue to give me hell, but maybe the next generation of educators will not be afraid of challenges to sacred cows or artificial canons. You seem bothered to have alternative models available, like Athenians afraid Socrates will "corrupt the youth". I'm not claiming your models are "wrong", only that they are not the only way to model. You guys, on the other hand, claim there's a natural uniqueness that points just to your sacred models. My approach is more open and more democratic: let multiple models peacefully exist and let model users freely choose. You are trying to spam bomb them away from my models, scaring them with Holy Model talk. If my models fail objective and relevant criteria, then let them fail. So far, I only see word-play attacks and dodgy claims about programmer WetWare. -t
[I've on occasion given you formal proofs. You completely failed to recognize them as such (Not surprisingly since you seem to think they need numbered clauses). Not that they are necessary to "kick your ass". BTW, if we were afraid of your challenges, we'd have you banned. Instead, we've asked you questions about your stuff. Questions you generally refuse to answer for one lame reason or another.]
No, you have not given such proofs. Sometimes there are issues over definitions in such alleged proofs, and you insist your interpretation is the center of the universe without evidence beyond ArgumentFromAuthority. And I have not consciously avoided any questions. Perhaps you worded them awkwardly; you guys are poor writers. Note that some have tried to get me banned in the past. You guys should thank me for helping you test and improve your poor writing/teaching/presentation skills so that you can rely less on ArgumentFromAuthority. I honestly believe that your heavy education has made you mistake the map for the territory, and it's my job as a relative outsider to slap you out of your slumber.
- {Most people, having understood a little bit about a subject, would realise they still have much to learn. Top is the only person I know who, having learned a little bit about a subject, realises only that he still has much to teach.}
- If you were so smart, you'd be able to produce clean objective logical proofs that didn't have to rely on vague wiggle-words. Instead, you mistake your PERSONAL definitions for universal truths, and then throw a tizzy when somebody challenges your vocab usage. Your arrogance writes checks your logic cannot cover. My favorite crap claim is that "thin" tables are objectively better. Does over-education really fuck up one's brain that much? "Science" should be removed from ComputerScience until fools like you stop abusing or ignoring science.
- {You've just provided more evidence in favour of my point.}
- Likewise. You have no specifics and no solid proofs.
- {Specifics and solid proofs have been given repeatedly. There's little point in presenting specifics or proofs, because you ignore them and argue around them. The closest I've ever seen is when you were given irrefutable evidence that prepared statements are more secure than ad-hoc queries, and even then you argued ludicrously and vociferously before going silent. I don't know if the obviousness of the truth finally got through to you and you were so humiliated by defeat that you couldn't bring yourself to publicly agree, or if you didn't agree but could think of nothing to defend more ludicrousness.}
- Bull. You fail to create ItemizedClearLogic, and insist your verbose round-about tangled writing style is "good enough". I think you are afraid of the scrutiny ItemizedClearLogic would entail, and that's why you make excuses and defend your goofy writing style.
[Very nice Top, you contradict your first statement with your second. (If I had never given such proofs, there could never have been issues over the definitions.) You've also provided more evidence of your ignorance on what a formal proof is. Definitions in a formal proofs are not interpreted. As for the bans, I've defended you from those banning attempts. As for the rest, feel free to show any instance of my use of ArgumentFromAuthority. I suspect you won't though. It's also ironic that you accuse me of mistaking the map for the territory, when a good deal of this page is you trying to change a question about natural numbers to a question about the notations we use when speaking about them.]
I meant good proof, not any ol' lame attempt. Lameness is easy to come by. Anyhow, let's see your best formal proof. Link it. Let's revisit your powerful genius and masterful clarity in action (cough cough cough). And humans have not found a decent way to test "notions" without using some kind of notation. You can't just claim that thing A has the equivalent semantics of thing B without some way to represent and compare semantics and/or it's equivalence so all can see; and such usually requires notation.
[They were good proofs. You just did what you usually do when evidence is presented against you. You try to change the subject. Just like you are doing here.]
Projection.
Bullshit! Link one about a key subject, or admit failure. I deny changing the subject for any key issues. (I do abandon irrelevant side BS, like I should do with that foo()=x claim.)
[Most of this page consists of you trying to change the subject. And all that was asked of you was to support one of your own claims. Now quit dodging and answer the questions.]
The prime analogy doesn't appear to correspond to our original semantics debate. I'm not going to waste time debating red herrings.
[It's neither an analogy nor a RedHerring.]

THANKYOU Top

Drifting Off Topic?

What does this topic have to do with language semantics being measurable or objective? If there's no connection, I should probably abandon my participation; it's growing into a LaynesLaw snake-pit. We have tests for primes that almost everybody agrees with (regardless of whether they are subjective or not), but we don't have the same kind of tests for "semantics". It smells like a false equivalency is being set up here. -t

[It's a counter to one of your arguments. One of your arguments is that semantics can't be objective because it's a general notion. The bit about primes shows that you can talk objectively about general notions, and in fact, humans have been doing so for thousands of years.]

You are putting words in my mouth. I did NOT say it "can't be", and I did not say semantics "are" a notation. They may be able to be proven objective, but YOU have not done such so far. And one can only talk "objectively" about them within the framework of notation. Math allows us to create fake worlds, UsefulLies, and we can make objective statements within those worlds using their made up rules. The utility of a UsefulLie does not prove objectivity.

[You did indeed say, "we can only say concrete things about explicit implementations, not general concepts." This is twice now you've been caught lying about what you've said on this very page. Now quit dodging and answer the questions.]

That's not the same as your "can't be" phrase. And "concrete" is not necessarily the same as "objective". In a specific model/notation, concrete things can be said, but they are not necessarily globally objective. One can say concrete things about imaginary numbers in the imaginary model, but that model may not fit a particular external phenomena. They can be concrete relative to a model. Primes can be concrete relative to a mathematical model/notation. I don't dispute that.

[What does "globally objective" mean, and if you didn't mean "objective" when you said "concrete", what did you mean by "concrete"?]

"Objective relative to some model/notation/assumption(s)" for lack of a better description at this time. Primes are "concrete" relative to the "usual" math models we use.
[Which of the two phrases did you just define?]
"Concrete". It's only meant as a working definition, and I'm not certain I used it consistently in the past.
[OK. With this definition, we can indeed say concrete things about general concepts.]
Be my guest...
[Sure. There is an infinitude of primes. Since this is objective relative to the standard model, it's objective relative to some model. It is thereby, "concrete". Since it is about primes, it's about a general concept. Therefore, it is something "concrete" we can say about a general concept.]
The "concrete" thing you can say about the general concept is that "There is an infinitude of primes in the model called 'standard'." That doesn't tell us anything new. It's a victory in technicality at best.
[Your definition doesn't require it to be new. In fact, it's better that it isn't new since we are all familiar with it.]
Familiarity is great, all things being equal, but we cannot scientifically analyze ideas/feelings/notions locked away in heads in any detail.
[It's a good thing we aren't talking about what's going on inside heads then. (BTW, there are whole branches of science devoted to analyzing, in detail, what goes on in peoples heads. I seriously doubt it's as impossible as you are imagining.)]
Much of what you talk about appears to be from a head model, not anything objective outside of a notation. For example, there's no evidence "integers" objectively exist in nature. They are a notion in human heads, and we create notations based around these head notions of integers, which are only useful if communicating parties agree on how to apply and interpret such notations. And whether such inside-head studies exist is moot if you are not using them.
[If there is no evidence that it exists in nature, then there is no evidence that it exists in your head. Your head is part of nature. We also aren't talking about communication here either, so the requirements on notation imposed by that need are also immaterial. I only brought up those studies since you were making a claim about how they couldn't be done to dismiss the current issues.]
Yes, heads are part of nature, but if you imagine a ghost, that doesn't mean the ghost actually exists, per common understanding. Life may indeed be nothing but a long dream ("row row row your boat..."), but we generally agree (by convention) that imagining that things are "outside" of one's head does not by itself make the those things "exist in reality". And what "couldn't be done" claim are you referring? I think you mistook "hasn't been done" for "couldn't be done". One day we may be able to scientifically study and analyze thoughts in detail. But that day is not here yet.
[There you go again, denying things that you've said on this page. You said, "cannot", and even is you really meant "has not", you would still be wrong. I've also denied that they exist physically, which is what is usually meant by "exist in reality".]
Sorry, what specific sentence are you referring to? Most "denial problems" are simply reference problems.
[In the response you start with "Familiarity is great". I can't say I've ever seen a denial problem be simply a reference problem. This certainly isn't one.]
No no no. I mean something like "statement 123 contradicts statement 456" or the equivalent. One thing I liked about classic BASIC is that every statement had an easy-to-find (required) line number.
[How does this relate to what you were responding to?]
Arrrrg. Explicit addressing of 2+ statements that allegedly contradict.
[How does that relate to what you were responding to?]
I give up. I dont know what the fucking goddam helll you are talking about.
[I'm asking you to explain how your "No no no." response relates to what I said immediately before it. You responded to what I said, but there doesn't appear to be a connection between them.]
I need a reference to TWWWWWWOOOOOOOO statements, not just one.
[The first statement was the one I was responding to. (Duh!) The second was the one starting with "Familiarity is great". (Duh! again)]
Arrrrg * 2

Anyhow, I don't want to get into a general debate about the philosophy of math at the moment. There are probably many things we commonly agree on regardless of whether they are "objective" or not such that starting from square-one is probably a distraction from the original issue. There's no reason to debate the objective existence of integers, for example, because I accept their existence as a working assumption. Thus, I ask to see if we can all re-focus on the issue of measuring semantics for equivalency (or similarity) for Algol-influenced/style dynamic languages. --top

[That's fine, you can come back to this topic whenever you feel ready to answer the questions. I will point out that the topic you wish to refocus us on is off-topic on this page. If you wish to discuss that, return to the page (or, even better, create a page devoted to that topic).]

(Moved from above)

{Furthermore, what is your definition of "subjective"?}

Depends on point of view or perspective.