## Circularity in Formal Languages?

For the discussion of math. Duh.

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Forest Goose
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### Re: Circularity in Formal Languages?

Dopefish wrote:(There exists forum magic that makes twerking the byproduct of the term for small adjustments to things, just in case you weren't aware.)

I am, I just couldn't help but smile when I read that -- my response to that wasn't related to the rest of my reply, just sharing my amusement.
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.

z4lis
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### Re: Circularity in Formal Languages?

The major change in realising that axioms don't exist; is in the way we understand the world around us. Again, we haven't removed a previous understanding - we have simply realised that part of our attempt at understanding was an illusion with no substance. No doubt this is one of the choke points for many people. The perception of having worked with axioms is compelling. Compelling to the point that mathematicians have ignored the evidence that has been available for at least a hundred years, in favour of believing in the illusion of axioms.

I honestly have no idea what you're talking about. I'm going to write down the axioms for "twerk theory", which studies a mathematical structure called twerks.

A twerk is a triple of sets (A,B,C) and maps f:A X B --> C and g:C X B --> A such that g(f(a,b),b) = a and f(g(c,b),b) = c for all a in A, b in B, and c in C.

That's it. Those are the axioms for a twerk. What fallacy have I committed so far? If I haven't done anything wrong yet, let me know and I'll continue talking about twerk theory until I run across the fallacy.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

Sizik
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### Re: Circularity in Formal Languages?

Formal Mathematics is a game.

You have symbols, and rules for how these symbols are allowed to be written out next to each other.

In one sense, Axioms are just legal starting spots, and Derivation rules are the rules about how you can chain together stuff.

As it happens, it is convincing that if we start with Axioms whose interpretation is true, and we pick Derivation rules that preserve truth, we can play this game and generate other formal statements, interpret them, and get true interpretations.

The fun thing is you can play this game with really formal mathematics, or less formal mathematics. So long as you are reasonable careful with interpretation, the "seed" statements (axioms), and the derivation rules as being property-preserving, you can play in this abstract game space and generate useful information outside of it.

A really basic example of this game is counting. An allowed interpretation of the number 7 is that it talks about 7 of something, addition as bringing together, and subtraction as taking away.

So if we have two groups of sheep, and one of them aligns with the interpretation of 7, and the other with the interpretation of 3, we can combine them (add) to get a group of 10 sheep. Really basic grade school stuff, but it is the game of mathematics.

Instead of having to think about the sheep individually (the one with the floppy ear, lame leg, grey coat, etc), we can map to numbers, play with numbers, then map back to sheep. I know I can break those 10 sheep up into 5 groups of 2.

Now, once my interpretation breaks down -- maybe when you combine the two groups, the sheep don't like each other, so one of the sheep is killed -- then the game is less useful, or I need to play a more complex game.

Calculus is just this game writ larger. If we have things that behave reasonably like smooth functions, we can do calculus on the smooth functions and get out results that can be aligned with the corresponding combination of the things. You can even describe how far from said smooth functions the things can be such that the result is still useful (so calculus when reality is "actually" granular on the Planck scale remains useful).

As it happens, this playing of games is fun in and of itself. We take systems, combine simplify or otherwise modify them, and see what results. We find some thing we wouldn't mind knowing, and try to figure out what game system makes it knowable. We find properties (like primes) of useful systems (like counting numbers) and play with them to see if anything falls out (like cryptography).

These games that interest us tend to be about things that are easy to reason about in a very vague way. When we play games on raw chaos, we reason about what patterns fall out. Even most extremely abstract mathematics is about the spots where there is structure (patterns we can understand) to sink our teeth into.

Such abstractions end up being useful more often than one might expect, even if we started without any practical application. Because once we find some neat properties, we can search for things for which this game corresponds to (or force it), and get an explosion of information from the interpretation of relatively basic things into our mathematical game.

But the axioms are not the foundation.

If we have things we correspond to a game (with axioms and derivation rules), and we find some neat things that are true in the game (and hence true about the things), if it turns out the game we are playing isn't a good game (it is inconsistent, for example), that doesn't cost us the truths we have discovered about the things that we verified independently of the game.

We even build games on top of other games (calculus on top of set theory, for example). The calculus game does not depend on the set theory game, despite being able to derive the calculus game from set theory.

Our knowledge of physics doesn't depend on the calculus game being a good game. The calculus game gave us a way to spread truth. If it is a bad game, some of the ways it spreads truth are not valid. But we check the game against physics: all it might do is provide us with clues where to look, and it is useful. If our particular calculus game turns out to be junk, we can invent a new game and play physics with it.

The axioms are not foundational, the game and correspondence between games is.

As you go through a mathematical and scientific education, you are taught various games. How to take equations and balance them, how to solve for x, etc. There are low level axiom systems with derivation rules that can be used to derive those games and their rules using formal mathematics, but that low level game isn't the reason why the higher level game works.

The game of solving for x has independent usefulness without the axioms of set theory. And you where taught that game and how to play it. You where not taught axioms and the like, because those really don't help you play the game of solving for x.

Now, formal axiom type math games have uses. They reduce what things you have to believe to be true in order for the interpretation game and derivation game to produce convincing results. So you can reduce more complex games to such "simpler" games and convince yourself those more complex games are sound, then continue playing the more complex games without having to go back to the low level game.

At one point, there was a hope to find the game to end all games. The one universal game from which you can derive the play all other good games. Kurt Gödel proved, by playing a game, that no game is universal: formal logic systems are either weak, incomplete or inconsistent.

Which means we have to invent useful games when we find a use for them, or just because it is fun.
she/they
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Treatid
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### Re: Circularity in Formal Languages?

It would seem that people are being blinded by the baggage certain symbols carry.

Let us substitute in some more neutral symbols:

for any A, A = B iff all C = B
all C are A
All C are in D
All D != B

You can see that the problem is structural. The specific meaning of the words is irrelevant. A never equals B.

For those who want to see an actual contradiction: "A statement has been defined": An A = B.

It doesn't matter what you think a 'definition' or 'meaning' is/are. It doesn't matter what you think a statement is.

It also doesn't matter how many levels of indirection you insert.

This structure goes nowhere. It does nothing.

The symbol 'meaning' has two* distinct meanings. There is the axiomatic 'meaning' and the informal 'meaning'. These two meanings are defined to be different. As much as some similarity or overlap might be desired (or even actually exist) - they are specified to not be equal to each other. Informal meaning is not the same object as axiomatic meaning.

If you want informal meaning to interact with axiomatic meaning - you must define that relationship. Axiomatic mathematics simultaneously says that meaning is specifically not informal - and then uses informal meaning as if it was axiomatic meaning.

*'meaning' is a symbol. We can map anything to that symbol. 'meaning' can have an unconstrained number of different meanings. The label is just a label, it is not the territory.

arbiteroftruth wrote:
Treatid wrote:2. A statement only has meaning with respect to a (defined) set of axioms.

False. Informal language has meaning. Not rigorous, "axiomatic" meaning, but meaning nonetheless.

a. I specified that 2. was an axiom. The question of truth doesn't apply to axioms themselves. Axioms need to be consistent, they don't need to be a-priori 'true' however that might be specified. We are using standard rules of mathematics to show that the standard rules of mathematics don't work. Standard rules say that axioms don't have to be justified/true. They just have to be consistent in context.

b. A symbol may have any number of different meanings. The informal meaning of 'meaning' and the axiomatic meaning of 'meaning' are defined to be distinct. Changing from one to the other half way through your chain of logic is only going to lead to confusion.

c. Your point is an illustration of the problem with axiomatic mathematics. Any given symbol can have an unlimited number of meanings. We have no way of specifying which meaning is intended. The labels we use are just labels. We can only manipulate the labels. We can't (that we know of) directly manipulate the concepts behind those labels.

Having said that - I'm attempting to show that an axiomatic approach to knowledge doesn't work; cannot work. But I think there are approaches to knowledge that do work. We can't have an absolute idea of what 'meaning' means. We can have a sufficiently agreed perception of what 'meaning' is to communicate - and to refine any given conception.

Agreed.

False doesn't apply to the second premise (axiom). Moreover, the apparent untruth only arises because you conflate two distinct uses of the same symbol.

Where does this leave us?

Forest Goose wrote:^
5 isn't true either. Every axiom of ZFC is in the language of set theory, which is a formal language.

Who are you trying to fool? Me or yourself?

You know that (in theory) the axioms of any axiomatic system can be expressed in terms of any other (sufficiently flexible) axiomatic system, in addition to informal language(s). Specifically, The axioms of ZFC have been expressed in multiple languages including informal languages.

You can, rightly, point out that, at some point, we need to use informal terms to get the process started

You directly concede that this is just indirection in any case - that sooner or later we end up back with the starting point based on informal language. So what is the point?

That we have to use informal terms to say things like what a string of symbols are, or that conjunction is commutative, or etc. doesn't really seem a problem.

Except for the fact that it creates a contradiction within axioms. Which means that every statement of axioms can be contradicted. Which means that axioms are a null system.

But other than that - no biggie...

2 is false too - this isn't the time of Euclid. Statements in formal languages don't have "meanings", that's not really how they work - the best sense of meaning I can see existing in axiomatic mathematics is in terms of models, and, in that sense, one major part of foundations is that ZFC has tons and tons of models. Heck, there's a model of ZFC + ~Con(ZFC), even; have fun with that:-)

See above.

It doesn't matter what the specific concepts involved are. It is the structure that doesn't work. The axiomatic approach to anything, of any kind whatsoever, cannot work. It doesn't do anything. The practical use of axioms has A=B alongside A!=B.

Either informal meaning is equal to axiomatic meaning; or it isn't. Similar, related, even same-ballpark doesn't cut it. The definitions say that they are explicitly not the same thing.

Maybe some progress could be made if the distinction between the two meanings of 'meaning' could be quantified. However, right now, the construction of axioms requires equality - not similarity.

4: neither are formal ones. However, theories in informal languages do use the axiomatic method, so there's that - heck, you're using the axiomatic method as we speak

Yes - I am using the existing concepts of mathematics to show that (some of) the existing concepts of mathematics are mistaken.

The argument has come up before that if I use axiomatic logic to show that axiomatic logic doesn't work - then my proof also doesn't work.

So... what next? The proof doesn't work - axiomatic logic pops back into existence. Ah - but then the disproof also pops back into existence. Obviously axiomatic logic works half the time - it oscillates between true and false - I guess you just have to catch it in alternate states?

By stating 3, you're giving yourself some problems, friend: one of the biggest bits of informal language is defining what a statement is - since you are, apparently, okay with statements (and, clearly, with using logic), I'm confused as to what you're disagreeing with...or, what, are you reductio'ing logic and language, supposedly? If that's how I'm supposed to read what you're doing, then my reply to you is "Flugsnatz,,,;goob-li=li-gock". Why? Because I declare my string defeats yours, and with logic and language gone, I can do so. In other words, you're either hanging yourself or admitting you reject everything, in which case anything goes - have fun on the horns of that dilemma

See above again.

Language hasn't disappeared. Just axiomatic mathematics. Axiomatic mathematics does take logic with it. Fortunately, reason (based on consistency) is still hanging around.

We could count before we had the formal number line. We can still count now that the formal number line never existed.

Axioms don't take away anything by never existing - except for some illusions. We very much want an alternate approach to knowledge. But losing an approach to knowledge that never actually did anything doesn't take away computers, bridges and buildings. Programming languages don't stop working and massive income inequality is still a thing.

The idea of infinity and the hierarchy of infinities are well shafted. But without any real world infinities, this won't make much difference to most people's daily lives.

Yakk: This has nothing to do with the intentions of axioms. I'm not asking that axioms be justified. I'm not making the mistake of thinking axioms need to be 'true'.

The problem is that axioms are a fiction. There is nothing there. There is nothing to work with. Every axiomatic statement ever made equals 'null'.

Informal language statements are not null. But axiomatic statements are defined to be null (rather - they are defined to not have whatever property axioms are supposed to imbue).

Go up top. Look at that structure. What are axioms adding that wasn't there before?

z4lis
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### Re: Circularity in Formal Languages?

I think you mistook me for Yakk. What the axioms are adding is a definition of what I'm calling a twerk. And those axioms are definitely there, since I've typed them.

Anyway, let's come up with three sets. How about A = {a,b} B = {1,2} and C = {p,q} and lets define:

f(a,1) = p
f(a,2) = p
f(b,1) = q
f(b,2) = q
g(p,1) = a
g(p,2) = a
g(q,1) = b
g(q,2) = b

And now if y in something in B, it's easy to check that

g(f(a,y),y) = g(p,y) = a
g(f(b,y),y) = g(q,y) = b
f(g(p,y),y) = f(a,y) = p
f(g(q,y),y) = f(b,y) = q

Given what I've just said, do you object to me calling the collection A,B,C,f,g a twerk? If you do, why? What fallacy have I committed? Can you give me any examples of a logical contradiction that could arise from actions such as these?

Also, "not making the mistake of thinking axioms are true" indicates you're totally missing the entire point of axioms. At no point in my axiomatic foundations of twerk theory did I assert twerks existed, and it doesn't even make sense to think that those axioms are "true" or "false", since I'm just giving you a framework for what I'm talking about when I'm discussing twerks. It's entirely possible that there are no twerks at all, but I have just provided one for you. There are lots more, however, and I'll give some constructions of new twerks from old using the axioms once you give me the logical green light to continue discussing them.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

arbiteroftruth
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### Re: Circularity in Formal Languages?

Treatid wrote:a. I specified that 2. was an axiom. The question of truth doesn't apply to axioms themselves. Axioms need to be consistent, they don't need to be a-priori 'true' however that might be specified. We are using standard rules of mathematics to show that the standard rules of mathematics don't work. Standard rules say that axioms don't have to be justified/true. They just have to be consistent in context.

Formal Mathematics is a game.

You have symbols, and rules for how these symbols are allowed to be written out next to each other.

In one sense, Axioms are just legal starting spots, and Derivation rules are the rules about how you can chain together stuff.

As it happens, it is convincing that if we start with Axioms whose interpretation is true, and we pick Derivation rules that preserve truth, we can play this game and generate other formal statements, interpret them, and get true interpretations.

In other words, no, axioms don't have to be 'true' within the context of pure mathematics. But math is useful because we can (informally) draw a correspondence between A: real-world truths with informal meanings, and B: mathematical statements with no meaning. When we do that, the mathematical game of deriving new statements from old statements corresponds to deriving new real-world truths from old real-world truths. That is how argumentation is done.

You can say your axioms don't need to be true in order for the logic of your argument to be valid, but in order for that to demonstrate anything (informally) meaningful, the correspondence between your axioms and real-world concepts needs to map the axioms to real-world statements that are true (in the informal context of real-world meaning).

Your argument is like saying physics is Newtonian because you're able to write down the equations for Newtonian physics. Yes, you can write down the equations, and they can be manipulated to generate new results, and we can agree on the logical validity of that process. But that doesn't tell me anything about the real world unless I also (informally) accept that those equations correspond to physical truths in the real world. As it happens, we know (informally) that those equations don't map to physical truths and are only a good approximation in certain circumstances.

Likewise, you can make an argument about A's being B's and B's being C's, and show that no A's are B's given the axioms, and we can agree on the logical validity of that process. But that doesn't tell me anything about the actual state and structure of mathematics unless I also (informally) accept that those axioms correspond to certain real-world truths about how mathematics is structured. As it happens, mathematicians know (informally) that not all statements used in math have to be defined with respect to axioms, and thus your argument does not apply to the actual structure of mathematics.

As for your argument about conflating two different uses of meaning; fine, let's be explicit in the distinction between rigorous meaning and informal meaning.

Then your first premise becomes: A mathematical statement only has rigorous meaning with respect to a set of axioms which themselves have rigorous meaning.

Then I reject that premise. At the very bottom, we have mathematical statements that are considered rigorous even though they are only defined with respect to "axioms" that have informal meaning and not rigorous meaning.

alessandro95
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### Re: Circularity in Formal Languages?

It seems that you ignored my post, can you please elaborate on how a network of relationships is anything that can be described by language, but is very different from an axiomatic system, despite this last one being describable by language?
The primary reason Bourbaki stopped writing books was the realization that Lang was one single person.

Forest Goose
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### Re: Circularity in Formal Languages?

Nope, sorry, I don't need informal language to state the axioms of ZFC. Also, you fail at reasoning. Your assumption was that I was required to use informal language to state them, not that they could be so stated, but didn't require it - so, even if I absolutely accept that informal language can state them, your premise still fails.

Nope, sorry, not indirection - you seem to be confusing axioms with formal logic, which aren't the same thing. Moreover, you seem to miss the point that if things like "Statement" cannot be defined formally using informal language, then we have the same problem using statements informally. Your argument is "Formal things have no meaning because they are defined informally" and "Informal things have meaning because they are defined informally". If informal methods can't define that a statement is a string of symbols, why can they do anything?

First, you're mixing levels: do you mean to say that using informal language to define a string of symbols contradicts axiomatic methods, somehow, or do you mean to say that it contradicts an axiom in every system of axioms? You seem to mix levels a lot, it makes you look kind of backward and incapable.

Second, you are, again, arguing against formal logic, not axioms, they aren't the same thing - this is another example of mixing levels.

Third, you use language like most people faking being smarter than they are, stop doing that, it makes it hard to even sort of take you seriously. For example, "...axioms are a null system", you keep using that phrase "null system" as if it were an important meaningful phrase that had a specific definition, but it is really just your way of asserting (falsely and without proof) that you think axioms are bunkum. I've noticed that a lot in your writing: phrases repeated over and over, like you use copy/paste to write responses, or you are trying to sound like you're saying more than some goofy opinion of yours.

So, what you mean to say is that you have no idea what model theory is?

See, my point was that you are the one going on about meaning in axioms, that's a central part of your argument. I wasn't responding, "Oh my, Treatid, but look, there is meaning!". I was saying, "Hey, dumbass, meaning isn't even part of the equation, this is the closest thing, notice how it goes against everything you're assuming anyway". That you are fighting against a concept from the time of Euclid is your problem, I guess, seems like anything more modern goes over your head.

I was trying to highlight that you have no idea what the distinction between axioms and formal logic are. As for your proof, it doesn't work in either situation, so no worries about things popping in and out of truth, your "proof" is just the bogus ramblings of a semi-educated lunatic upset that C = P = NP = T doesn't work so well.

...you've been arguing against the latter, not the former the entire time.

You also didn't respond to my point, or, more likely, didn't understand it.

You are either using a reductio against logic and language or you are arguing against them within them. The former is absurd for the reasons I pointed out, the latter doesn't work by it's own premises. See, that's the problem, you're both accepting everything you're arguing against and rejecting it - the real problem is that you don't, unshockingly, understand the topics involved and are confused. You've made up some artificial division, then are trying to save the side of the line that is appealing to you while rejecting the side that isn't.

The problem is that you have no idea how to do that, no clue what characterizes either side, and no ability to reason...notice how you just keep stating the conclusions in question like they were already agreed upon? Notice how you keep asserting nothing, saying gibberish, then reasserting nothing? Notice how you have yet to present the asked for examples, arguments, basic proofs of competency, etc.? Yeah, as I've mentioned, we aren't having an argument, you're just foaming at the mouth.
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.

Gwydion
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### Re: Circularity in Formal Languages?

Treatid wrote:1. Any contradiction in an axiomatic system means that every statement of that system can be contradicted. Such a system is null.
2. A statement only has meaning with respect to a (defined) set of axioms.
3. Axioms are, themselves, statements.
4. Informal languages (and the elements thereof) are not defined in the axiomatic sense.
5. Initial axioms are expressed using informal language(s).
Treatid wrote:I specified that 2. was an axiom. The question of truth doesn't apply to axioms themselves. Axioms need to be consistent, they don't need to be a-priori 'true' however that might be specified. We are using standard rules of mathematics to show that the standard rules of mathematics don't work. Standard rules say that axioms don't have to be justified/true. They just have to be consistent in context.

Treatid, you claim that axioms are statements (3). These axioms thus only have meaning with respect to a defined set of axioms (2). However, the axioms are expressed using informal language (5) and informal languages are not defined in an axiomatic sense (4). As a result, your axioms are not formally defined which contradicts the preceding sentence. By (1), your system is null.

The question most of us are asking is, why should we care? You've defined an inconsistent system, but are the only person here arguing that system actually represents axiomatic mathematics as we understand it. If you found a contradiction in one axiomatic system, that doesn't unravel all of math, just the one inconsistent system - in this case, yours.

jestingrabbit
Factoids are just Datas that haven't grown up yet
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### Re: Circularity in Formal Languages?

Seriously, when are we going to get to model theory? This stuff is boring.

((By which I mean:

2. A statement only has meaning with respect to a (defined) set of axioms.

is actually false. The modern methodology works not just with axiom systems but with systems that model those axioms. In this way we can come up with quite different structures that model the same set of axioms, with some statements true in some model and false in others and vice versa. This allows us to interrogate the amount of "give" in a system. We also have systems of axioms that are "complete" where all statements are decidable without reference to a model, which can actually be quite interesting really (the field axioms, for instance).))
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Forest Goose
Posts: 377
Joined: Sat May 18, 2013 9:27 am UTC

### Re: Circularity in Formal Languages?

Setting aside my enjoyment at mocking you, I will say this in sincerity, hopefully you will take my advice and rethink your launching into screeds.

Your arguments, and variations thereof, are something I've seen countless times - mathematics attracts lots of railing against formalism and searching for philosophical loopholes. There is a reason for this, it's because mathematics is challenging in a way that no other subject is (not even the mathematics heavy parts of physics can be): mathematics allows for no, and I mean no, winging it, no giving the gist, no loose arguing by analogies, no playing it a little loose with the terms, etc., and, moreover, things get dense quickly.

A complete novice can stumble around on the web, read some articles about this, that, and the other thing pertaining to philosophical theories, or political notions, or literature, or, to some extent, physics, biology, etc. And, then, proceed to pontificate, or, at least, argue for some of their own ideas and thoughts - this isn't to say that those ideas will be good ideas, but there is enough wiggle room to give the person the impression they are reasonably discussing the topic (especially if the entire group is lay people). However, with mathematics, you cannot do this, it just doesn't work - a group of interested folk might read a few popular science books about worm holes, then discuss them over drinks; a group of interested folk will have no such luck discussing number theory, or zfc, or the application of schemes in logic. This isn't because mathematics is the hardest discipline of all, but because it is all technical, all academic; there's nothing outside of the formal elements to appeal to, no foot-half-in-the-door place from which to casually toss about ideas.

Thus, long story short, mathematics is intellectually terrifying for a lot of people, because while you can bullshit about your interpretation of Billy Joe's dream in literary classic X, or tell people, and sound impressive doing so, about the neat properties of particle Y, or gene Z, or technological advancement K, you can't do this with math, you just sound dumb, and obviously so.

Then, there's that second point: math is dense; but also interesting. For example, lot's of people think Fermat's Last Theorem is nifty, however, there is no pretty picture that conveys why it is true - if you want to have an idea of how it all works, best grab a few books on horizontal iwasawa theory, galois cohomology, schemes, modular forms, etc. etc. etc. The Millennium Problems are interesting, right? Want to talk about P-vs-NP, the most intelligible of the bunch, time to start discussing oracles, and relativising proofs, and reductions, and etc.. As for the others on that list, like the Hodge Conjecture, you need to study for a good while, a real good while, to even be able to understand the question being asked.

In other words, mathematics has captured a lot of lay people's attention, but it lacks a way for them to feel they understand the topic (you can convince yourself you get the gist of particle physics, even if you don't, etc.). So, putting this together, there's plenty of people interested in problems X, Y, and Z, but that feel threatened and challenged by everything they encounter actually regarding them.

Thus, some people respond by objecting to the whole enterprise of mathematics, rather than admit that, just maybe, they aren't quite as smart as they thought they were - a couple afternoons on Wikipedia will not convince you, nor your friends, that you have a clue about group cohomology; but it will let you pretentiously ramble about type/token distinctions (even when you don't really get them).

You, in my opinion, are one of these people. You claimed, in the past, that you are interested, or understand, computer programming - I'm guessing you are somewhat educated about various tech topics - you have, clearly, read some popsci books on physics, you (I'm guessing) scroll around and drop comments in a philosophy forum, or two, and, most likely, look up academic topics in Wikipedia (or something else of that sort) - and, generally, I bet, you feel fairly comfortable with what you read and feel that you can reasonably comment on these various topics without being challenged every three words. Does that sound right so far?

However, going with the usual, at some point you got similarly intrigued in some math problem, P=NP, or something, and found that when you tried to advance your ideas that, not only where they rejected, everyone responded as if you were incompetent - as if you were speaking gibberish. Given your feeling of comfort when you talk about "blag blah bloz epistemology" or "ra ra ra C++", and a more accepting response, your initial gut reaction is that math folk must be some exclusive club of bullshitters and egotistical folk who are just going on with their crooked dogma - trying to look smart and belittle those not in the club. Thus, what's the next step? Puff up the ego, inflate your sense of "I'm bright enough", and, then, wade on in and confront them - expose their bullshit, as you said, show "the emperor has no clothes", etc.

The problem, as mentioned, is that this just isn't accurate - a few random physics books can let you think you know physics, despite not knowing it; a few philosophy articles can convince you that you know the subject, despite not; etc. In other words, it feels like you hit a wall with math, one you didn't expect, and that's troubling enough to put up a fight; whereas with other subjects, you peek over the wall, pretending that you've passed over it. (I'm not saying you can't learn anything, or don't know anything, but most people know a single subject well and think they know various other subjects reasonably well - despite that not being true at all.)

So, my advice: drop this line of absurd reasoning and ranting. Instead, pick up some math books, work through the problems, ask questions, be humble, etc. etc. etc. Or, on the other hand, move on to something else that you want to invest that level of effort in (maybe you don't want to spend 10,000 hours learning what the Hodge Conjecture is asking, after all). At any rate, please, I beg of you, quit following the route of the math crank, the man who is unable to fathom that he cannot, and does not, know everything; that way leads to nothing, everyone will eventually see you for what you are, it's nothing but an inevitable stumbling around to various communities and groups, spouting off as long as they will tolerate you, then, assuming them fools and narrow-minded, moving on. Please, don't do this, nothing good waits at the end of your current path, trust me, nothing is there save shame, misery, and perpetual blows to your ego.
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.

gmalivuk
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### Re: Circularity in Formal Languages?

And on that high note, /thread
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
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