## The Largest Number We Will Ever Need

• 7.6k
No

Why? Well, given how math is supposedly an axiomatized system, there can be no issues with the conclusions that follow among which number the irrationals. However, that means we could play around with the axioms to disallow irrationals, oui? Is this possible/no?
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My favorites are cranks who say that mathematical logic, even just sentential logic, is all wrong, as they are typing and reading on computers that are packed to the panels with Boolean programming and would not even exist if not for the invention of modern digital computers in the very crucible of mathematical logic.
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we could play around with the axioms to disallow irrationals, oui?

Si.

I've told you about a million times already, you can have axioms for whatever you want*, even inconsistency if that's your thing.

* 'ceptin sometimes you can't have all of what you want at the same time, as we found out from Godel.
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Many years ago I was watching the Academy Awards (don't ask me why I would waste my time that way). Whoopi Goldberg was at the microphone, and she said something like, "We don't need the space program. We should get rid of it. Nothing ever came out of the space program", as she was being broadcast via satellite. You gotta love it.
• 7.6k
Many years ago I was watching the Academy Awards (don't ask me why I would waste my time that way). Whoopi Goldberg was at the microphone, and she said something like, "We don't need the space program. We should get rid of it. Nothing ever came out of the space program", as she was being broadcast via satellite. You gotta love it.

:rofl:

To be fair she could've been talking about our interplanetary ambitions, moon landing and all that jazz. Frankly speaking, she has a point - all that money earmarked for space ventures could be spent on more pressing matters of which, if the news is to be believed, there are many. Perhaps it's an escape plan. :chin:
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inconsistency

Gödel? If it is consistent, it is incomplete; if it is complete, it is inconsistent.

@javi2541997 $\uparrow$
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If it is consistent, it is not complete; if it is complete, it is inconsistent.

If it is recursively axiomatized and has enough of arithmetic, then if it is consistent then it is incomplete (which is to say that if it is complete then it is inconsistent).
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all that money earmarked for space ventures could be spent on more pressing matters

What could possibly be more pressing than people getting their satellite dishes on the roof to watch reality TV shows of people eating bugs?
• 2.2k
Gödel? If it is consistent, it is incomplete; if it is complete, it is inconsistent.

Good quote and reference! :up:

It remembers me about shûnya (empty) on Buddhist metaphysics.

Relative Existence or No Self Nature: Nothing has a essence, nature, or character by itself. Things in isolation are , shûnya, "empty." The nature of things only exists in relation to everything else that exists. Existence as we know it is thus completely relative and conditioned by everything else.
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Just like you can't do very much math without relations. But with just one relation* you can do it all.

* membership
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Yet, in line with the Buddha's thoughts, we're all alone. The relations we build are, some say, fragile to the point of being mere illusions (the imagery of rats fleeing a sinking ship is enough to send shivers down me spine). That, perhaps, is what Buddha meant - the ties that bind us are maya, no self (anatta).
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What could possibly be more pressing than people getting their satellite dishes on the roof to watch reality TV shows of people eating bugs?

:lol:
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pi is not a ratio of two rational numbers.

pi is the ratio of the circumference of any circle and its diameter. But if the diameter is rational then the circumference is not. So still pi is not the ratio of two rational numbers.

So there is not a contradiction.

The measurement of one is incommensurable with the measurement of the other, therefore the relation between the two measurements is an irrational ratio. So I wouldn't really call it a contradiction, it's just an attempt to do the impossible, to establish a relation where one cannot be properly established due to that incommensurability. A straight, one dimensional line is incommensurable with a curved two dimensional line. Likewise, an attempt to give the cardinality of an infinite set is an attempt to do the impossible.

Since music is numinous in nature, it being somewhat of a bridge between us and the universe, the Pythagoreans probably extrapolated the math found therein to the universe itself.

This was primitive wave theory.

The discovery of irrationals, kind courtesy of Hippasus of Metapontum who was thrown overboard to prevent word of this getting out, threatened to overturn what was up to that point a perfect world. A simple and yet magnificent way mathematics could serve as the foundation of the universe had to be abandoned. I wonder what Max Tegmark has to say about this?

The temporal nature of waves always seems to throw a wrench into the cogs of the application of mathematics toward understanding the foundation of the universe.

I know, I know, pay no attention to the man behind the curtain.

Step out from behind that curtain. On with the show this is it!

Mathematicians come up with general formulas, involving pi and other irrational numbers. Isn't it the case that engineers make use of those general formulas, from which they can decide what specific specific values to use as close enough for the task at hand?

Yes sir! But what happens when understanding the foundation of the universe is "the task at hand"?
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Yes sir! But what happens when understanding the foundation of the universe is "the task at hand"?

Good luck with that. :roll:
• 99
Sadly, it seems math discussions on TPF are doomed to descend to the level of farce.

Absolutely correct: the level of crank on this site is ridiculous

Thus, the diagonal of the square is now commensurable with its side, which we've known to be untrue since the time of the Pythagoreans!

Interestingly enough, Avicenna's argument against (mereological) atomism was that applying the Pythagorean theorem is empirically successful, and it could not have been had our physical space been akin to a taxicab geometry, something like what the atomists suggested (in fact, in a taxicab geometry, using the Pythogrean theorem would not even approximate our values!). This later on came to be known as the distance function argument
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Good luck with that.

Thanks, but unlike the undisciplined mathematicians who make willy nilly axioms however they please, we don't rely on luck here.
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Thanks, but unlike the undisciplined mathematicians who make willy nilly axioms however they please, we don't rely on luck here

Luck's role in your life is inversely proportional to the quality of your plans.

:snicker:
• 289
... unlike the undisciplined mathematicians who make willy nilly axioms however they please ...

Oh Great Anonymous Forum Poster - who clearly knows better than some of the greatest minds humanity has produced for the past 5000 years - enlighten us : please give an example of these wicked axioms that mislead our youth that we might know them and revile them.

(This should be good.)
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I didn't say the axioms are wicked, I said they are wlly nilly. And it was Tones who stated that idea. I just went on to draw the conclusion that when you create your principles in such an undisciplined way, you need luck in the application of them. Here:

I've told you about a million times already, you can have axioms for whatever you want*, even inconsistency if that's your thing.
• 1.7k
One can propose whatever axioms one wants to propose. That doesn't entail that mathematicians propose axioms willy nilly or in an undisciplined way. Indeed mathematicians very carefully propose axioms on various criteria.

That one can do X, or is permitted to do X, doesn't entail that one does X.
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I reject your productivity and outcomes argument, whatever rubric it correctly falls under.
— TonesInDeepFreeze

[...] your actions fall short of moral excellence (in that indignancy fail prudence and justice)
Kuro

[...] in not acting rightly one retracts from the virtue of their character, which is less so a property of any particular fault or flaw in action rather than those habits which become second nature to them and come to form their decisions & methodology (which is what the discussion came about, the generalized implications versus the particular one, hence the relevance of the quote).Kuro

If I understand, your view is that by doing things that are not right, one become habitual in doing things that are not right, thus harming one's character.

My point is that I don't agree that it is not right to decry the egregiousness of cranks.

Here you apply an outcomes/productivity argument:

I find myself as frequently frustrated as Mr. Tones with respect to the mathematical, logical or other formal/technical errors that are somewhat frequent on this forum. However, a rude attitude seldom yields anything productive: there is the option of politely leaving a discussion, perhaps at no fault of your own but the inadequacy of your interlocutor, or explaining their mistake at a reasonable level.

"Fanning the flame", so to speak, is unnecessary in whole. [...]
Kuro

So, if I understand, you base the claim that the act is not right, and thus harms character, on the claim that the act causes bad outcomes and is unproductive. Also, you think the act is wrong because it is acting from indignancy which is not prudent and just.

(1) I think that the act has a better outcome than not decrying the egregiousness of cranks, and that it is productive. Not productive toward having bonhomie with the crank, but productive in another way.

(2) I don't think it is wrong for posters to post for reasons other than certain outcomes or productivity, and I don't limit my own reasons that way.

(3) I don't view virtue ethics as determining, not even the main determinant. I see that virtue ethics, in my limited understanding, offers a lot, but I don't see that it should be the sole, or even main, framework to be used alongside a balance among other frameworks.

(4) I don't think it's incumbent to so seriously apply ethics, let alone pretty demanding ethical frameworks, to all aspects of posting. There is a lot of posting that I don't think has good outcomes or is productive that I wouldn't think of cudgeling with application of an ethical theory or even think of objecting to it at all. Posters don't ordinarily think "Will this post have good outcomes, will it be productive, is it free of any breach of virtue that will harm my character?" and it would be nuts for me or anyone to expect they would. People usually post at their pleasure, to express themselves, to advocate for their ideas*. And that expression may include voicing of indignation.

* Though there are other motives too, such as learning from other posters, enjoying pleasant agreements (rare, in this forum), etc.

(5) Perhaps I don't know your context, but I don't think that expressing indignation is necessarily imprudent or unjust.
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I didn't say the axioms are wicked, I said they are wlly nilly.

Fine, willy nilly then. You miss the point : somehow the truths of math have been revealed to no one else but you.

I ask that you stop being so selfish and share this revealed folk wisdom with the world. Begin here by detailing some misbegotten axioms you have encountered. Then write up your math musings and send them off to prestigious math journals. I am sure they will fight to be first to publish. Your fame and fortune await.
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$The worst crank on this forum claimed that I lack training in philosophy. (1) I did not say I have not studied philosophy. In philosophy, I am not a scholar, and I have not retained many of the particulars I learned a long time ago, but I have taken courses in philosophy (and not just philosophy of mathematics) and read books. In the philosophy of mathematics, I am not a scholar, but I have read many books and articles. In mathematical logic and set theory, I'm not a scholar, but I have a good handle on the basics through taking courses and careful study of several textbooks, and I have also compiled a rigorous log of formalizations, definitions and proofs in set theory and the first stages of some other branches of mathematics. Meanwhile, I don't opine in threads on various philosophical discussions where the level of conversation would require me to be more adequately versed. And whatever particulars I've forgotten about Western philosophy through history, I still retain much of the understanding and appreciation of philosophical methods themselves. Philosophy of mathematics may be discussed at a general level that does not reference particular mathematical developments, but usually, as in this forum, discussions in the philosophy of mathematics do turn the mathematics itself. In that regard, it is crucial that that mathematics not be mistaken or misrepresented. Indeed, threads in this forum are often premised as critiques of set theory and classical mathematics. Critiquing set theory and classical mathematics is great and vital. But the critiques need to be based on actually knowing what is being critiqued. A critique from ignorance, confusion, and misrepresentation is intellectual and philosophical garbage. So what about the crank himself? There is not a hint in what he writes that he knows even the least thing about such basic contexts of philosophy of mathematics as Frege, logicism, Hilbert's formalism, Godel's 'What Is The Continuum Hypothesis?' constructivism, intuitionism, reverse mathematics or anything else really. The crank himself knows not a thing about the first order predicate calculus, second order logic, formal axiomatics, set theory or, as far as I can tell, any mathematics beyond everyday arithmetic. But that doesn't stop the crank from claiming it's all nonsense. Indeed, the crank insults mathematics itself and mathematicians themselves. Of course, the crank does not fault himself for lack of training in philosophy of mathematics, though he spews his ignorance, illogic and literal nonsense about it regularly, for years. And what would the crank say about other posters who we can tell are sorely "lacking in training" in the philosophy of mathematics? What is the training in philosophy of mathematics of the original poster of this thread? Let alone what are his philosophical commitments? Does the crank (and another poster) hold that everyone must have a philosophical commitment to qualify for posting? I don't believe posters must. * The original poster started with a presumably empirical question, then went on top claim or insinuate that infinitistic set theory is wrong, and it is still not clear whether the original poster has clear position as to whether he's an ultrafinitist, even that he is a particular kind of finitist. The crank pounced on my humble statement that I don't hold to a particular philosophy, saying that I "lack training" in philosophy. The crank shows his pettiness, illogic, and factual incorrectness. And lies about me. (2) It is not necessarily a fault not to have a philosophy. One can be open minded about many philosophies and employ their virtues. /$ The crank says it is annoying for a philosopher to have to see a non-philosopher post in a philosophy of mathematics discussion and insist that philosophers shouldn't discuss the metaphysical bases on which mathematical axioms stand, if they have not first studied mathematics.

(3) The crank shouldn't presume to speak for others.

(4) The crank is lying about me. I have never said that one must study mathematics to talk philosophically about mathematics. What I have said is that one should not spew disinformation and confusion about the mathematics without knowing anything about it. The posts by cranks are not just philosophy but they include claims about the mathematics that is the subject of the philosophy. I don't even say that one shouldn't talk about that mathematics without first studying it; rather that one should not misrepresent it, and that the first step to not misrepresenting it is to learn at least a little bit about it.

\$ The crank says that it is philosophy that is being discussed not mathematics.

(5) The cranks lies again. The profuse record of actual posts show that the cranks make many claims about the mathematics and that the discussions reference the mathematics. This very thread is one of them.

The crank himself, at nearly the start of this thread wrote: "Try naming pi to its final decimal place."

The crank lies, directly belying his own posting, when he says the discussions are about philosophy but not also the mathematics itself. How can that be topped?
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In the philosophy of mathematics, I am not a scholar, but I have read many books and articles. In mathematical logic and set theory, I'm not a scholar, but I have a good handle on the basics through taking courses and careful study of several textbooks

Hey, I'm a fan. You have my respect. :cool:

I used to pick up bits of modern math knowledge from @fishfry before he departed.
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I really appreciate that. It means more to me than you might think. Thank you, jgill.
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somehow the truths of math have been revealed to no one else but you.

Really, where did you get that idea? I am in the habit of dismissing what are commonly touted as the "truths of math", for being in some way faulty. How do you get from this to the point of saying that the real "truths of math" have been revealed to me.
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That was quite the rant Tones. I hope you're feeling better now, to have gotten that off your chest. And if so, I'm very happy to have been able to assist you, in feeling better about yourself: if that was even possible. If it wasn't possible, for you to feel better about yourself, as I fear is the case, then I'm sorry, for you.
• 289

So there are no truths of math? What, in your wisdom, is math after all?

I assume that - other than simple arithmetic with positive integers (basically, what you can do with your fingers and toes) - you believe mathematics to be made up gobbledy-gook. To live in the modern world and be such a willful know-nothing is breath-taking.

I begin to discern what happened : someone received a low grade in calculus at university and has had it out for those idiot mathematicians ever since.
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What, in your wisdom, is math after all?

Math is based on made up axioms, like Tones described. It's imagination, fiction, not truth. The majority of the axioms which get accepted into the mainstream do so because they prove to be useful. Some though, may be accepted simply for beauty or eloquence. Usefulness is quite distinct from truthfulness.
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I said that one my choose whatever axioms one wants. That does not imply that there are not many mathematicians who choose axioms on the basis of their truth. Among mathematicians and philosophers of mathematics, there are different notions of how axioms ought to be chosen.
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