2+2=4 is less controversial. But if I put up a thread that read "Is 2+2 really the same as 4", we can plainly see that I'm not asking whether current mathematics teaches that or what calculators output as an answer to that question. And 0.9999... deals with infinity and limits making it more complex. But when I programmed mathematics into a turning machine, I had to hard-wire certain mathematical operations into it like xor, or, and. Maybe our brains are hard-wired to believe certain self-evident axioms that are just not true. How we perceive the world is based off of what's best for our survival as a species, not necessarily what's true. In a "2+2" thread I would break down the thought process that goes into convincing someone that it is the same as 4. The law of identity and the law of non-contradiction would come up.wtf wrote:Excerpt from a conversation I recently had in my head:

Imaginary You: Thank you so much, wtf, for taking the time and trouble to write out that clear explanation of the mathematical argument that .999... = 1. It's better than anything in my dark and fuzzy real analysis book, and way more understandable than anything my real analysis professor ever said.

Me: You're very welcome.

I wish to talk about one thing at a time to focus this conversation.marsh8472 wrote:Correct, I was thinking 1 was not also a representation of a real number when I said that. The point though is if any property distinguishes the left side from the right side and we include that property as part of the criteria for the equal sign then they are not the same. Such as "0.999..." has ellipsis in it, has 9's in it, is 8 characters in length, represents an sum of an infinite series..wtf wrote:So humor me. Talk about the law of identity with respect to 2 + 2 = 4, and how it does or does not fall victim to the exact same criticism you leveled at .999... = 1.

I ask you directly, are you questioning 2 + 2 = 4 on the same grounds that you question .999... = 1? Because the same argument applies. Different symbols, a plus sign, etc. So they are not the same.

Let me argue as you might that 2 + 2 is not necessarily 4.

* Isn't it the case that in math, "2", "+", "=", and "4" are symbols that have historically contingent definitions?

* And isn't it the case that the modern set-theoretic definitions are arguably nonsense? For example we are told in set theory that ∅ is the empty set; and that we define:

0 = ∅

1 = {1}

2 = {0, 1}

3 = {0, 1, 2}

4 = {0, 1, 2, 3}

etc. This definition is due to John von Neumann. https://en.wikipedia.org/wiki/Set-theor ... n_ordinals

* And as I say, isn't this definition manifestly nonsense that any schoolboy would recognize as such? For one thing, there is no such thing as the empty set. It's a mathematical fiction whose purpose is to make everything else work out; but any philosopher worth his or her salt could rip the concept to shreds with ease.

* Isn't it completely wrong to claim that the counting numbers are "sets," when in fact mathematical sets are not actually common sense collections as we're told in school, but that are actually very weird hypothetical abstract entities that, when you drill down into the technical bits, aren't anything like any collections we ever saw?

* Didn't philosopher Paul Benacerraf write a brilliant paper called What Numbers Could Not Be, in which he did in fact rip to shreds the notion that numbers are sets? http://isites.harvard.edu/fs/docs/icb.t ... cerraf.pdf

* And while we're on the subject, isn't it therefore the case that the set-theoretic statement, "3 is an element of 4" true is set theory, and obviously and profoundly false in the real world, in common sense, and in fact even in mathematics, outside of the field of set theory?

* And aren't these set-theoretic definitions used to prove in modern math that 2 + 2 = 4?

* Therefore I ask you: Isn't it entirely possible that someday mathematicians will throw off this nonsense, provide a far better definition of the natural numbers, and then it will be taught that 2 + 2 ≠ 4? After all, Pluto.

What say you to that argument? I say it's no different than .999... = 1, and that the only reason .999... gets so much attention is because it involves mysterious limits, whereas 2 + 2 = 4 seems to be an everyday fact of nature.

But arguing from "everyday facts of nature" can be misleading. It was considered an everyday fact of nature for thousands of years that the geometry of space was Euclidean. Kant even claimed that humans have an a priori knowledge of this fact. In the 1840's when Riemann and others demonstrated the logical consistency of non-Euclidean geometry, his discovery was met with at first shock, and then with the dismissal that it was no more than a mathematical curiosity that had nothing to do with the real world. And the some guy named Einstein came along and showed that the weird mathematics of non-Euclidean geometry was in fact true about the world.

You made some other points in your post that I'll get to but let's do one thing at a time. The 2 + 2 = 4 example has all the features of .999... without the counterintuitive baggage, so it's instructive.

tl;dr: Do you deny 2 + 2 = 4 on exactly the same grounds that you deny .999... = 1?

Hardly solved unless you use that term loosely. There's a difference between having one solution and having it solved. They have one particular answer which is 0.9999... = 1. Other possible answers are 0.9999... < 1 or maybe 0.9999... is undefined like 1/0. Mathematics returns to us whatever assumptions we feed to it. What's the smallest number greater than 0? Well there is no such number n the real numbers, which has no holes in it, has no smallest number greater than 0. Does that make sense to you? If I walk from point A to point B, did I walk across an infinite number of points or a finite number of points? Real numbers provides one way to comprehend it but no real way to verify it outside of itself.wtf wrote:Yes, it was a challenging question for two thousand years before Newton and for another 200 years after. And determining the value of infinite sums is certainly a challenging problem for freshman calculus students.

But it's a solved problem. It's no objection to a scientific or mathematical fact that it used to be unsolved. If it's solved today, it's solved. It's sophistry (there's that word again but what other word would you use?) to claim that because a solved problem used to be unsolved that this somehow casts doubt on the solution. Fire used to be an unsolved problem. The wheel used to be an unsolved problem. The internal combustion engine, heavier-than-air flight, and television used to be unsolved problems. Today they're solved. Philosophers don't say, "Oh well, fire might not actually be hot, because after all it used to be an unsolved problem." Come on, man.

And maybe they'll decided that it's better that a knight move a different way if there's a good justification for doing it. Like rules change in many games such as NFL football. http://operations.nfl.com/the-rules/evo ... nfl-rules/wtf wrote:But outside of math class it's not clear what the question even means. It's far from clear that it means anything at all.

If I'm teaching you chess and I say the knight moves in such-and-so manner, isn't it a category error to say, "Well yes, in chess class the knight moves that way. But how does it REALLY move?"

extremefinitism did make some good points at the end though. Mathematics uses some misleading language. Like when they say x approaches infinity. Any finite number is no closer to infinity than any other, whether it be 1 or 10^googleplex, so how can x approach infinity? It's probably best to say 0.9999... has a variable number of 9's instead of infinite 9's when we talk about the reals. A variable number of 9's is less than 1. They don't have infinitesimals or "infinity 9's" in the real number system.wtf wrote:I'm painfully familiar with that site. He's a complete crank. He doesn't even know what finitism is.

It so happens that finitism and its even more extreme cousin ultra-finitism are two extremely interesting philosophies of math. I myself am of the opinion that ultra-finitism is the only philosophy of math that makes sense in the real world and has even a potential claim to be true. The problem with it is that it's not axiomatizable. But neither is physics, and that doesn't stop anyone from doing physics.

If you want to read something intelligentabout finitism and ultra-finitism, have a look at this. https://plato.stanford.edu/entries/phil ... thematics/. Also this is pretty good. http://math.stackexchange.com/questions ... believe-it. And Wiki's not half bad. https://en.wikipedia.org/wiki/Ultrafinitism

I'd love to talk about ultra-finitism, it's super interesting and actually very technical. There was a serious, accomplished professional mathematician at Princeton named Ed Nelson who was the world's leading advocate of ultra-finitism. He passed away just a couple of years ago.

But listen man you have simply got to stop randomly linking crank sites. I have no idea what your point is in doing that. It makes me think you're not intellectually serious about any of this.

ps -- Serious alternative philosophies of math: Intuitionism/constructivism. Finitism/ultrafinitism. Structuralism (today actually winning the war and supplanting set theory. Benacerraf's article I linked got that party started).

Not serious: Internet crank sites.