Part 2

I will remind you, that Pythagoras demonstrated the irrational nature of the square. The relation between two perpendicular sides of a square produces the infinite, which as I argued above is bad. This makes the square a truly impossible, or irrational figure. And, all "powers" are fundamentally derived from the square. Therefore any exponentiation is fundamentally unsound in relation to a spatial representation.. — Metaphysician Undercover

A lot here to work with. Let me break it down a little at a time. I've already made many of these points in other threads lately so I'll try to keep this short.

I will remind you, that Pythagoras demonstrated the irrational nature of the square. — Metaphysician Undercover

Yes. This is true. Euclid's version of the proof, from around 300BC, shows that there are no two integers whose ratio squared is 2.

It shows that the square of the ratio of two integers is never 2. This is a result in number theory. It's indisputable.

That it comes up so naturally, as the diagonal of a unit square, shows that (at least some) irrational numbers are inevitable. Even if we're not Platonists we suspect that a Martian mathematician would discover the irrationality of $\sqrt 2$. There's a certain universality to math. Euclid's proof is as compelling today as it was millenia ago. The philosopher has to acknowledge and account for the beauty, simplicity, and power of Euclid's proof, undiminished for 2400 years; not just deny mathematics because some number doesn't happen to be rational.

The relation between two perpendicular sides of a square produces the infinite ... — Metaphysician Undercover

No. No no no no no no and no. I've hit this point several times in other recent threads, and already enumerated these bullet points over in the other thread. Let me just recapitulate these points briefly.

* A real number is not its decimal representation. $\sqrt 2$ happens to have an infinite, nonrepeating decimal representation. That's an artifact of decimal notation, not a characteristic of the real number $\sqrt 2$ itself. There are many finite characterizations of $\sqrt 2$.

* $\sqrt 2$ is a computable real number in the sense of Turing 1936. Therefore even a diehard constructive mathematician would gladly accept the existence of $\sqrt 2$. From now on when I say existence I mean mathematical existence. I'll stipulate that we can't measure $\sqrt 2$ in the real world. As long as you'll stipulate that we can't measure 1 either. So the real world doesn't even come into play here. Measurement's approximate. Mathematical numbers are exact, but not necessarily aspects of the real world. I gather this is much of your thesis anyway. I agree.

* $\sqrt 2$ has a repeating continued fraction representation. I won't explain what that is but the Wiki article's pretty good.

Now the continued fraction representation of $\sqrt 2$ is $[1; 2, 2, 2, 2, ...]$, or "1; followed by all 2's." You can't get any more finite that that. That phrase completely characterizes $\sqrt 2$ as a continued fraction. There's no priority between decimals and continued fractions. It's just that one's taught in high school and the other in math major number theory class. They're both equally valid ways of representing a real number.


3) Euclid's proof depends only on Peano arithmetic (PA). In terms of proof strength, PA is the same as ZF without the axiom of infinity. So a diehard finitest would accept the irrationality of $\sqrt 2$. A finitist accepts the existence of each of 0, 1, 2, 3, ..., but not a completed set of them. You don't need a completed set of natural numbers to prove that the square root of 2 is irrational. It's a finitary fact, not an infinitary one.

So far we've seen that both constructivists and finitists would have no problem accepting $\sqrt 2$. If we informally rank real numbers based on believablity, the noncomputables are the most unblievable reals. Those are the numbers that genuinely encode an infinite amount of information. Next in believability are transcendentals like $\pi$. They're computable, but they're mysterious. Their existence wasn't even proven till the 1840's.

The most believable irrationals are simple algebraic numbers like $\sqrt 2$. Numbers that are the roots, or zeros, of polynomials with integer coefficients like $x^2 - 2$. Those are super-easy to construct without using infinitary methods; and as the quadratic formula from high school shows, the ones that are the roots of quadratic polynomials are easy to determine.





* $\sqrt 2$ lives in a finite extension of the rationals. This is the abstract algebraic approach I've talked about recently in another thread.

But there's a much more down to earth way to explain this point. The idea is to view our ever-expanding number systems as the solution to the problem of solving some type of equation.

So in the beginning (again this is thematic, not necessarily historical) we have the positive integers 1, 2, 3, ... We can solve equations like $x + 2 = 3$ with no problem, the answer is $1$.

But what if we try to solve $x + 3 = 2$? We're stuck. We haven't got a number to solve this equation. So we invent one. In fact we invent zero and all the negative integers just so we can now solve more equations.

Now later on this invention gets formalized into the abstract algebra that I've presented. But the essential idea is just solving as many equations as we can, even if we have to make up numbers to do it. And over time, everyone comes to believe in these made up numbers.

So ok, now someone gives us the equation $5 x = 7$. Now we're stuck, there's no integer, positive or negative, that will work. So we have to invent the rational number $\frac{7}{5}$ to solve the equation.

So @Meta you see that the rational numbers, which you think are so sacred, are themselves just made-up fictions that at one time were controversial and not widely believed in.

Ok now we have the rationals. What do we do with an equation like $x^2 - 2 = 0$? We have to invent $\sqrt 2$; and by extension all of the other real numbers of the form $a + b \sqrt 2$ where $a, b \in \mathbb Q$.

Now of course we need to solve $x^2 = 3$ and all the other quadratics; and we have to solve all the third degree equations, and so forth. There is a class of real numbers called the algebraic numbers that contains all the solutions of all the polynomial equations.

It's tricky though because what about the simple quadratic $x^2 + 1 = 0$? We have to invent a new kind of number, the imaginary unit $i$ and the complex numbers, the set of all $a + b i$ where $a$ and $b$ are real.

And this process keeps going. There are quaternions and octonians and sedonians and more. There are weird numbers like the p-adics. There are Cantor's tranfinite ordinals and cardinals. There are matrices and tensors, beloved by the physicists.

Every time mathematicians need a new kind of number they invent one; and then after a passage of time, people come to believe in these new numbers.

There was a time when people believes in the integers but not the rationals. You believe in the rationals but not the irrationals. But there's nothing special about the rationals. They were once a fiction too. Your choice is arbitrary and not based on anything but the era in which you were born, and your level of mathematical education.

which as I argued above is bad. — Metaphysician Undercover

But no. It's not bad. It just is. And remember, if you lived in a different age you'd think negative numbers were bad. Then you'd think zero is bad. Then you'd think rational numbers are bad.

Your feeling about what numbers are real and which aren't is purely a matter of accident based on the age you live in and your level of mathematical knowledge.

I hope you can get outside yourself to get this point. There's nothing special about the rational/irrational jump. It's one jump in a long historical line of mathematical sophistication from putting a mark on a cave wall every time you kill a mastodon; to doing the most advanced research mathematics and phyiscs. You can't draw a line that says this step is real and the next step isn't. The rationals are just as fake -- or just as real -- as the irrationals.

So it's not "bad" that $\sqrt 2$ is irrational. It just is. Some numbers are rational and some are integers and some are p-adics and some are quaternions. Mathematicans have lots and lots of numbers. They all have mathematical existence.

This makes the square a truly impossible ... — Metaphysician Undercover

Of course a unit square is not impossible, it's the most obvious geometric figure there is after the equalateral triangle. Speaking of which, an equalateral triangle with sides of 1 has an altitude of $\frac{\sqrt 3}{2}$. Can't avoid those pesky irrationals, they are literally all over the place. Everywhere you look.

But to say a square is "impossible" simply because you don't like the great discovery of Pythagoras ... that seems a little beyond the pale for serious discourse in my opinion.

, or irrational figure. — Metaphysician Undercover

Equivocating irrational as in "not a ratio [of integers]" with irrational as in cray-cray. Come on, man, you can do better than that.

If the sides of a square are rational the diagonals are irrational. Man get over it. It just is, like the sun rising in the east. A fact about the world. That's one of the curious things about abstractions. Even though we make them up in our minds, they still bear definite truth values. 5 is prime even if you don't believe in the Platonic existence of 5. As a philosopher of math you need to account for that, not deny it because you don't like the Pythagorean theorem.

And, all "powers" are fundamentally derived from the square. — Metaphysician Undercover

I'm not sure what you mean but from what follows I'm guessing you mean squares, cubes, hypercubes, and in general n-cubes; which, by the way, are perfectly well understood in math.

Therefore any exponentiation is fundamentally unsound in relation to a spatial representation.. — Metaphysician Undercover

That's just silly. There's nothing fundamentally unsound. The diagonal of a unit square can't be expressed as a ratio of integers. It's just a fact. It doesn't invalidate math. Even the Pythagoreans threw someone overboard then accepted the truth.

The tl;dr on all this is that you're just mistaken that there's anything special about rational numbers, other than the fact that (by definition) they're ratios of integers. Lots of naturally occuring mathematical constants turn out to be irrational.

Among the irrationals, the very simplest are the quadratic irrationals like $\sqrt 2$, meaning that they're roots of quadratic equations. They're so easily cooked up using basic algebra, or Turing machines, or continued fractions.

The bottom line is that your unfamiliarity with certain aspects of math is leading you to philosophical errors. $\sqrt 2$ has a very strong claim to mathematical existence and it's a finitary object, not an infinitary one. Likewise $\pi$ is computable, hence encodes only a finite amount of information. Decimal representation is not determinative of a real number's infinitary nature.

I've read through much if not most of your writings in this thread. I think I understand where you're coming from. I found two remarks I can get some traction on. — fishfry

Thank you fishfry. I'm vey impressed that you actually took the time to read and try to understand what I was saying. Most just dismiss me as incomprehensible or unreasonable and go on their way.

So ok. ZF[C] is unsound. This is a commonplace observation. Might you be elevating it to a status it doesn't deserve? Are you thinking of it as an endpoint of thinking? What if it's only the beginning of philosophical inquiry? — fishfry

As Aristotle explained, logic proceeds from the better known, toward understanding the lesser known. The premises are the better known, the conclusions lesser known. A conclusion requires multiple premises so if there's a possibility that each premise is incorrect, the probability is multiplied in the conclusion. Therefore the "endpoint" of thinking is more uncertain than the beginning point. The engineer applies mathematics as if there is a high degree of certainty in the axioms (premises), and uses these toward producing an understanding of what is until then unknown.

The principles we apply in any philosophical inquiry must be of the highest degree of certainty in order to give any credibility to the conclusions of the inquiry. Science and engineering apply mathematics with a confidence that the underlying axioms are sound. If the engineers thought that the mathematical axioms were unsound, they would request better ones. A philosopher such as myself may approach the axioms with skepticism. But the skepticism must be justified by underlying principles with higher certainty. If one were to cast doubt on any particular axiom, that philosopher must appeal to further principles which are known with an even higher degree of certainty. That's what I believe I am doing, referring to principles with a higher degree of certainty, such as the laws of identity and non-contradiction, to cast doubt on these mathematical axioms.

What Russell says is true to an extent, people who apply mathematics do not generally question the accuracy of the axioms. However, in the passage quoted he fails to mention the reason for this. The reason that they do not question the accuracy of the axioms is that they have confidence in those axioms. So yes, applying mathematics is no different from applying other logic, we proceed with the underlying knowledge that if the axions are true then so will be the conclusions, when the logic is properly applied. However, what is not mentioned in the passage is that when we apply mathematics we have a high degree of confidence in the truth of the premises, and that's why we are using mathematics.

That, I submit, is one of the key issues of the philosophy of math. We agree that math isn't true, but it nevertheless seems intimately related to the real or the actual.

This is what I mean when I say that "ZCF is unsound" is the starting point of philosophical inquiry into the nature of mathematics; not the end of it. I get the feeling you think it's the last word. It's only the first. — fishfry

Perhaps you and I can find an agreeable starting point for a philosophical inquiry, here. We agree that the axioms of mathematics are not true, so we can now examine the confidence with which mathematics is applied. Mathematics does not consist of "true principles", so the confidence is not based in truth. I suggest that the confidence is based in utility. It has worked in the past, it works today, therefore it will work in the future. Mathematics is reliable, therefore we have confidence in it.

Now, this reliability indicates that mathematics is, as you say "intimately related to the real or the actual". I have no problem agreeing with you on this. So we can ask, what produces this reliability, what is the nature of this relationship with the real or actual. I propose that the reliability is produced by approximating the truth. Reliability does not require that we know the absolute truth concerning the matters we are involved with, but it does require that we have an approximation of the truth, and this approximation lowers the probability of mistake, increasing reliability. The nearer we can approximate the truth, the higher will be the reliability. In this way we remove "truth" from the absolute logical categories of is/is not, placing it into the relative, degrees of probability. Propositions are not judged to have truth or falsity in an absolute way, they judged for probability of truth.

That's just silly. There's nothing fundamentally unsound. The diagonal of a unit square can't be expressed as a ratio of integers. It's just a fact. It doesn't invalidate math. Even the Pythagoreans threw someone overboard then accepted the truth. — fishfry

Didn't you just agree that in a very fundamental way, mathematical axioms are unsound? And you supported this with the quote from Russel. Or do you think it's just some axioms which are unsound? ZF is unsound but Euclidian geometry is sound. What about the parallel postulate? Anyway, if some axioms are sound and others unsound, we'd have to revisit all the axioms anyway, to distinguish the degree of soundness.

Sure, it's "a fact" that Euclidian geometry produces points which have a distance between them as immeasurable, but it's also a fact that this is a problem which has not been resolved. We've agreed not to speak in terms of soundness or truth of the axioms, so all we can look at is whether the axioms are problem free, and these geometrical premises create problems, i.e. that there are points with immeasurable distance between them.

That it comes up so naturally, as the diagonal of a unit square, shows that (at least some) irrational numbers are inevitable. Even if we're not Platonists we suspect that a Martian mathematician would discover the irrationality of 2–√2. There's a certain universality to math. Euclid's proof is as compelling today as it was millenia ago. The philosopher has to acknowledge and account for the beauty, simplicity, and power of Euclid's proof, undiminished for 2400 years; not just deny mathematics because some number doesn't happen to be rational. — fishfry

Let me apply the principle described above, to √2 now. Truth is relative, not absolute. First, it is probably not true that a square is natural. The right angle is artificial. It was produced for various purposes, surveying plots of land, establishing parallel lines, etc.. So it is highly unlikely that √2 comes up naturally, it comes up as a result of our desire to create parallel lines, or whatever other purposes we use the right angle for. Therefore, our desire to create parallel lines and such, has produce the right angle, which is extremely reliable for these purposes, so it must approximate the truth about "space" to a fairly high degree.

However, there is this problem which the right angle creates, and that is that it allows us to very easily make two points with an immeasurable distance between them. The "circle" and "pi" has a very similar problem, so this geometrical system based on a point with degrees of angles around the point is suspicious. Now, I see two ways we could go with this inquiry. We could follow principles like Peirce's and say that's simply the way space is, there's a vagueness about it which incapacitates us in this way, making it impossible to measure the distance between these points. The nature of space is such that we cannot determine points in space, only vague infinitesimals. That assigns "the fault" to the object, space. Or, we could proceed in the way that I recommend, and consider that our desire to make parallel lines and such, has misguided us relative to truth, such that we modeled space with right angles etc., which was an approximation to the truth, but maybe not accurate enough for the purposes we now have. This assigns "the fault" to the subject, the model. The problem being that the circle and the right angle do not properly represent the spatial area around the point. Notice that the latter way, which I propose, gives us inspiration to delve into our spatial models, divulge our past mistakes and produce a better approximation of space. The former way suggests that it is impossible to model space more accurately, because we cannot model a precise point. And that's just the way that space is, impossible to model in such a way as to rid ourselves of the conclusion that if locations are represented as precise points, there will be points with an immeasurable distance between them. So we are uninspired to look for that better way of modelling space. Therefore, if we are inspired to increase the degree of reliability and certainty with which we apply mathematics, we need to revisit this model of space, to see where it leads us astray.

Among the irrationals, the very simplest are the quadratic irrationals like 2–√2, meaning that they're roots of quadratic equations. They're so easily cooked up using basic algebra, or Turing machines, or continued fractions. — fishfry

That irrational numbers are "easily cooked up" from our axioms is clear evidence of weakness in the axioms. As explained, irrational numbers represent things which cannot be measured using the existing axioms. If more and more axioms need to be layered on in a seemingly endless process, to create the appearance that these immeasurable things actually can be measured, this is simply closing the barn door after the horse has run away.

However, there is this problem which the right angle creates, and that is that it allows us to very easily make two points with an immeasurable distance between them. — Metaphysician Undercover

It seems to me your entire post is deeply confused, rampant with ambiguity, amphiboly, conflation, undefined terms - or if they'e defined then the definitions are not held to - and faulty argument, all in a toxic mix and mess, that like most messes, is easy to make but labor intensive to clean up . Above, for example, is an example. You claim it is possible to "make two points with an immeasurable distance between them." I assume you mean there are many sets of two points in which the distance between the two points in each such set is "measurable." As opposed to the two points "easily made" you mention. You can make me go away to an "immeasurable distance" by making very clear how you might realize your claim. But I think you cannot, and if you cannot, then I expect you to acknowledge same. So this is a challenge to drive you off your ground, either to advance with a better argument, or withdraw.

I assume you mean there are many sets of two points in which the distance between the two points in each such set is "measurable." As opposed to the two points "easily made" you mention. You can make me go away to an "immeasurable distance" by making very clear how you might realize your claim. — tim wood

You seem to be lacking in reading skills tim, it's no wonder you're so confused. I was talking about what someone can do with Euclidean geometry, not about what someone can do walking on the ground. So your example is way off track.

If you want to argue that mathematical objects and geometrical constructs are not real objects, then I'm in agreement with you there, and that's the position I've taken on this thread. And, the fact that we can create an idealized object (created through the use of defined terms, rather than drawing, or walking on the ground), like a circle or a square, and make these objects such that there is immeasurable distances within the objects, is further evidence that these are not truly "objects". My argument has been, that a mathematical object, like what is referred to by "2" or "3" does not qualify as an object in any way consistent with the law of identity. And set theory is based in, and requires, this false premise that these are objects. In other words, set theory violates the law of identity.

If you're interested, go back and read the thread, Zuhair and I covered much ground. Fishfry was in and out, and for the most part did not keep up with the discussion, but now seems to have a renewed interest.

You seem to be lacking in reading skills tim, it's no wonder you're so confused. I was talking about what someone can do with Euclidean geometry, not about what someone can do walking on the ground. So your example is way off track. — Metaphysician Undercover

Geez, I tried to be explicit in my challenge to your statement. I presupposed Euclidean geometry, but a) maybe I should have been clearer, and b) what does it have to do with my challenge to you?

Here's what you wrote:

and that is that it allows us to very easily make two points with an immeasurable distance between them. — Metaphysician Undercover

Here's what I wrote:

You claim it is possible to "make two points with an immeasurable distance between them."... You can make me go away to an "immeasurable distance" by making very clear how you might realize your claim. But I think you cannot. — tim wood

And guess what - you'll never guess; it will be a complete surprise! You never even paid "the cold respect of a passing glance" to that which I challenged you to do.

On the assumption that there can be two points the distance between them being "measurable," how do you "very easily make two points with an immeasurable distance between them"?

On the assumption that there can be two points the distance between them being "measurable," how do you "very easily make two points with an immeasurable distance between them"? — tim wood

Sorry, I didn't understand your challenge, you said something obscure about making you go away. I'm afraid you have your own free will and I can't make you go away.

Now that you've made you question clear, I'll answer it. But it should be obvious from what I already wrote, so I don't understand why you're confused. Construct a square according to the rules of Euclidian geometry, the opposing corners are an immeasurable distance apart.

It seems to me your entire post is deeply confused, rampant with ambiguity, amphiboly, conflation, undefined terms - or if they'e defined then the definitions are not held to - and faulty argument, all in a toxic mix and mess, that like most messes, is easy to make but labor intensive to clean up . — tim wood

Thanks man. I was going to say the same but @Metaphysician Undercover and I have reached a point of mutual civility and I'm trying to keep that going. So I didn't say what you just did, though I endorse and agree with it.

I have been reading @Meta's post and it's hard to know where to start. @Meta, do you agree that math isn't physics? You complain that we can't measure $\sqrt 2$ but we can't measure 1 either. All physical measurement is approximate and never exact. You don't seem to appreciate this point.

When you say "squares are impossible" you seem to be denying abstraction. Of course there are no perfect right angles in the world, but there are in abstract reasoning. You can't deny abstractions, civilization runs on them.

I'm afraid you have your own free will and I can't make you go away. — Metaphysician Undercover

Hey it's great. @Meta is polite to me now and insulting others. Thanks @Tim for bearing this burden on my behalf.

Construct a square according to the rules of Euclidian geometry, the opposing corners are an immeasurable distance apart. — Metaphysician Undercover

Yes, you've got the diagonal of a square. Why is that distance "immeasurable"? Are you quite sure that "immeasurable" is what you mean? Because it seems simple enough to me to measure it.

Thank you fishfry. I'm vey impressed that you actually took the time to read and try to understand what I was saying. — Metaphysician Undercover

You're welcome.

Most just dismiss me as incomprehensible or unreasonable ... — Metaphysician Undercover

No, really?

Just kidding!! I enjoy the new civility and hope to perpetuate it.

I couldn't tell who was crazy, you or me. You were so sure of yourself when you said I didn't understand anything about anything. I wanted to find out what was going on. I believe I did. You think math should be physics. You think perfectly accurate distances exist in the real world. You seem to deny abstraction, as in your claim that squares are "impossible," your word. You did not yet take my point that rational numbers are just as questionable as irrationals. I mean, what the hell is $\frac{7}{5}$, anyway? You can't measure it. You can't show it to me. You can't evenly divide a pie into seven pieces. [Insert clever pun on pi here]. So in fact your psychological belief that rational numbers are more deserving of mathematical existence as irrational ones, is simply that: a psychological belief.

I have explained in recent posts (here and the other thread) that:

* The number $-4$ is a made-up mathematical gadget that allows us to solve the equation $x + 5 = 1$.

* $\frac{7}{5}$ is a made-up mathematical gadget that allows us to solve the equation $5 x - 7 = 0$.

* The number $\sqrt 2$ is a made-up mathematical gadget that allows us to solve the equation $x^2 - 2 = 0$.

* The imaginary unit $i$ is a made-up mathematical gadget that allows us to solve the equation $x^2 + 1 = 0$.

In each case, mathematicians of a given historical age become interested in an equation they can't solve. They say, "Well what if there were a funny gadget that solved the equation?" Then after a few decades the made-up gadget becomes commonplace and people come to believe in it as a first-rate mathematical object, and not just a convenient fiction. Eventually we can construct these gadgets both axiomatically in terms of their desired properties; and as explicit set-theoretic constructions. They do indeed become real. Mathematically real, of course. Nobody ever claims any of this stuff is physical. You are fighting a strawman.

Note also that new classes of numbers can be explained as algebraic phenomena, not just geometric ones. The diagonal of a square is one vision; the solution to an equation is another. The fact that we have multiple ways to arrive at the same place is a clue that we are dealing with truth. Not physical truth. Abstract mathematical truth. If the geometers hadn't discovered the square root of 2, the algebraists would have.

So this is where I want to engage. You seem to want math to be what it can never be. You want an abstract, symbolic system to be real, or actual. That can never be. Your hope can never be realized.

As far as the larger philosophical issue:

We agree math isn't literally true about the world. We wonder what its status is. Well, there are things that are physical and things that are abstract. Physical things are presumed true. That's a rock on the ground, if I pick it up and drop it, it accelerates toward the earth at 32 feet per second$^2$.

But abstractions can be real as well. Driving on the left or right is a social convention that varies by country. It's artificial. Made up. It's nothing but a shared agreement. It could easily be different. Yet it can be fatal to violate it. Driving laws are social conventions made real. See Searle, The Construction of Social Reality.

I submit that 5 is prime and the square root of 2 both exists and is irrational. Those are abstract truths. They are not physical. But they are true. [Typo fixed].

We can go further. "5 is prime" is not like traffic lights. In chess we can make up variants of the game where the pieces are arranged differently or there are pieces with different types of moves. Math is a formal game as well, but we are NOT FREE to say that 5 is not prime. This I think is the core mystery of math. 5 is prime even though there is no 5 and there are no primes.

I have been reading Meta's post and it's hard to know where to start. @Meta, do you agree that math isn't physics? You complain that we can't measure 2–√2 but we can't measure 1 either. — fishfry

I am satisfied with this principle if we can apply it consistently. We do not measure mathematical "objects", they are tools by which we measure objects. That's why I argued that they are not proper "objects". Now let's apply this to set theory. Cardinality, for example is a measure. If the applicable principle is that we do not measure mathematical "objects", then why allow this in set theory? It's inconsistency.

So either we can measure mathematical objects, like squares, and the sides of squares, just like we can measure the cardinality of sets, or we cannot measure these so-called mathematical objects. I'm fine with the latter principle so long as we maintain consistency. But if we allow that we can measure these so-called objects, then we can measure a square, and find that the diagonal cannot be measured.

Yes, you've got the diagonal of a square. Why is that distance "immeasurable"? — tim wood

It's what we call an "irrational number", implying an immeasurable length. Are you familiar with basic geometry?

I believe I did. You think math should be physics. You think perfectly accurate distances exist in the real world. You seem to deny abstraction, as in your claim that squares are "impossible," your word. — fishfry

This is not at all what I've been saying, so I think we might not really be making any progress.

In each case, mathematicians of a given historical age become interested in an equation they can't solve. They say, "Well what if there were a funny gadget that solved the equation?" Then after a few decades the made-up gadget becomes commonplace and people come to believe in it as a first-rate mathematical object, and not just a convenient fiction. Eventually we can construct these gadgets both axiomatically in terms of their desired properties; and as explicit set-theoretic constructions. They do indeed become real. Mathematically real, of course. Nobody ever claims any of this stuff is physical. You are fighting a strawman. — fishfry

Neither you nor I is talking about physical objects here. What we are talking about is the "made-up gadgets" which you describe here. You seem to imply that there is a difference between these funny gadgets, and "first-rate mathematical objects" I deny such a difference, claiming all mathematics consists of made-up gadgets, and there is no such thing as mathematical objects. But this is contrary to set theory which is based in the assumption of mathematical objects. If you really think that a "funny gadget" becomes a "mathematical object" through use, you'd have to demonstrate this process to me, to convince me that this is true.

So this is where I want to engage. You seem to want math to be what it can never be. You want an abstract, symbolic system to be real, or actual. That can never be. Your hope can never be realized. — fishfry

How can you not see that this is a problem for set theory? Set theory assumes that it is dealing with real, actual mathematical "objects". That is a fundamental premise. Now you agree with me, that mathematics can never give us this, real or actual things being represented by the symbols. So why don't you see that set theory is completely misguided?

But abstractions can be real as well. Driving on the left or right is a social convention that varies by country. It's artificial. Made up. It's nothing but a shared agreement. It could easily be different. Yet it can be fatal to violate it. Driving laws are social conventions made real. See Searle, The Construction of Social Reality. — fishfry

So your argument is that the "funny gadget" gets made into a "first-rate mathematical object" through convention, just like driving laws. But those are ";laws", not "objects". Let's suppose that the mathematical symbols referred to conventional laws instead of "objects", as this is what is implied by your statement. How would this affect set theory? Remember what I argued earlier in the thread, sometimes when a symbol like "2" or "3" is used, a different law is referred to, depending on the context.

I submit that 5 is prime and the square root of 2 both exists and is rational. — fishfry

I don't see how "the square root of 2 exists" could possibly be true, It is an irrational ratio which has never been resolved, just like pi. How can you assert that the solution to a problem which has not yet been resolved, "exists"? Isn't this just like saying that the highest number exists? But we know that there is not a highest number, we define "number" that way. Likewise, we know that pi, and the square root of two, will never be resolved, so why claim that the resolution to these problems of division "exist"?

I submit that 5 is prime and the square root of 2 both exists and is rational. . . . But they are true. — fishfry

What am I missing here? :chin:

Yes, you've got the diagonal of a square. Why is that distance "immeasurable"?
— tim wood

It's what we call an "irrational number", implying an immeasurable length. Are you familiar with basic geometry? — Metaphysician Undercover

Why is the diagonal of a square an "irrational" and "immeasurable" number? What you have completely missed is what you actually wrote. Of course the side and the diagonal are together incomeasurable. But either by itself is perfectly measurable. How might this be done?

For the side of the square, take a straightedge, lay it along the side and mark the straightedge at the ends of the side. That's the length of the side

For the diagonal, same procedure, giving the length of the diagonal. And what you and I and almost everyone else understands perfectly well is that the lengths marked on the two straightedges cannot both be expressed in the same system of measurement with rational numbers. But they both can be measured to the same arbitrary level of precision.

As pointed out repeatedly by others, the problem with irrationals is not with their number, but with some of their numeric representations. And as to the irrationality of diagonals by themselves, there is nothing at all to prevent anyone from constructing a square with an integer diagonal - in that case, of course, the sides would be irrational.

I submit that 5 is prime and the square root of 2 both exists and is rational. . . . But they are true.
— fishfry

What am I missing here? — jgill

I'm making the point that the facts of math are not quite as arbitrary as the facts of other formal games such as chess. That's something peculiar or interesting about math. We can't make up a form of math where 5 isn't a prime (in the usual integers). Yet 5 and "prime" are abstract and somewhat fictional entities.

I'm not entirely sure what your question was. But the point is that math isn't like driving laws or chess. Some things in math are objectively true yet not facts about the physical universe.

For the side of the square, take a straightedge, lay it along the side and mark the straightedge at the ends of the side. That's the length of the side — tim wood

The issue though, is that we are talking about measuring the ideal square, just like set theory talks about measuring the ideal numbers. We are not talking about measuring a representation of the ideal square, which is written on the paper, just like set theory is not concerned with measuring the numerals, it is concerned with "the numbers" represented by the numerals.

As pointed out repeatedly by others, the problem with irrationals is not with their number, but with some of their numeric representations. — tim wood

This is absolutely false. The reason why the "numeric representation" of irrational numbers is a problem is because there is no number to be represented. How it is proven that the square root of two is irrational is a demonstration that it breaks the rules of "rational numbers". Pythagoras proved that the square root of two is not a rational number. This means that if we're using the rational number system, the square root of two falls outside of that system, there is no number for it. In the other thread, fishfry called this "a hole" in the rational numbers, but I disagreed with that term.

In reality therefore, an irrational number, has a numeric representation, we clearly have a numeric representation of pi, and square root two, but there is no corresponding number for these representations. As fishfry indicated in the other thread, we might create a new number system (the real numbers) and try to include what is represented by irrationals, within that system, as numbers. I do not understand the construction of the real numbers, but I am willing to argue that this is a flawed approach. Instead of addressing the real problem, which is the fact that we can produce spatial representations (circles and squares) where numbers do not apply, and adjusting these representations accordingly, the mathematicians have created an extremely complex number system, which simply veils this problem.. In other words, instead of addressing the real problem, which is a feature of our faulty spatial representations, and trying to solve that problem, the mathematicians have just hidden it under layers of complexities.

When irrationals are identified as being rational-number generating algorithms, irrationals are obviously well-defined. But of course we are rarely interested in algorithms per-se, and we also like to use irrationals as numerals, when for instance we use them to label the length of the diagonal of a square.

Now obviously, our visual impression of a square is vague to a certain extent, as is any physical measurement of a square, and if we repeatedly measured the diagonal of an actual square with very high precision we would obtain a range of (rational valued) measurements. In other words, by labelling the diagonal of a square with, say, sqrt(2), we are using Sqrt(2) not as denoting an algorithm, but as denoting an arbitrary rational or distribution of rationals, that is yet to be determined through a process of (repeated) measurement.

This is another example of why game-theory is a good model of mathematics use.

I do not understand the construction of the real numbers, but I am willing to argue — Metaphysician Undercover

Don Quixote at least had dignity.

This is absolutely false. The reason why the "numeric representation" of irrational numbers is a problem is because there is no number to be represented. — Metaphysician Undercover

Are you saying there is no square root of two?

Are you saying there is no square root of two? — tim wood

There is no rational number which is equivalent to what is represented by "square root of two". So the problem is not as you say, one of numerical representation, it is that there is no number for what is represented. There is a numerical representation, commonly used, but no number which is represented by it.

There is no rational number which is equivalent what is represented by "square root of two". So the problem is not as you say, one of numerical representation, it is that there is no number for what is represented. There is a numerical representation, commonly used, but no number which is represented by it. — Metaphysician Undercover

You apparently take "rational number" to mean the same thing as "number." And you apparently think that rational numbers possess some quality that those irrational thingies don't and can't have. What quality might that be? What quality can you name, identify, describe that rational numbers share but that the irrationals cannot have - other than, of course, being rational.

And if you allow for there being such a thing as the square root of two, never mind what it might be beyond being the square root of two, then you're obliged to acknowledge that it has a lot of brothers and sisters, in fact a very large family. So large that in comparison, the rationals are next to just no size at all.

Perhaps it might help if you tell us what you suppose a number, or number itself, to be. No doubt the difficulties here originate in the foundations of your thinking.

I'm not entirely sure what your question was — fishfry

"I submit that 5 is prime and the square root of 2 both exists and is rational. . . . But they are true."

The square root of two is rational? Am I misreading your sentence? :worry:

The square root of two is rational? Am I misreading your sentence?/quote]

Oh that's a typo, sorry. Is that what you were asking earlier? Yes typo of course. I'll go back and fix it. — jgill

You apparently take "rational number" to mean the same thing as "number." And you apparently think that rational numbers possess some quality that those irrational thingies don't and can't have. What quality might that be? What quality can you name, identify, describe that rational numbers share but that the irrationals cannot have - other than, of course, being rational. — tim wood

That sums it up pretty well. Irrational ideas are incoherent, so there is good reason to rid our conceptions of such things. What we like in our conceptions is the quality of being rational and we are wise not to accept irrational ones conceptions.

And if you allow for there being such a thing as the square root of two, never mind what it might be beyond being the square root of two, then you're obliged to acknowledge that it has a lot of brothers and sisters, in fact a very large family. So large that in comparison, the rationals are next to just no size at all. — tim wood

No, I do not allow that there is such a thing as the square root of two, that's the whole point, and it is what I just explained to you in the last post. Saying "the square root of two" is really a matter of saying something, which represents nothing real, just like "pi" is a matter of saying something which represents nothing real. The mathematics clearly demonstrates that there is nothing real represented by these expressions. So it really doesn't make sense to say that there is "such a thing as the square root of two", just like it doesn't make sense to say that there is such a thing as pi. These are symbols which we use because they are extremely useful, but we ought to respect the fact that there really is nothing which is represented by them. You might imagine an ideal square with a diagonal line bisecting it, or an ideal circle with a line bisecting it through the middle, but the mathematics demonstrates that these are incoherent images.

Perhaps it might help if you tell us what you suppose a number, or number itself, to be. No doubt the difficulties here originate in the foundations of your thinking. — tim wood

A number is a definite unit of measurement, a definite quantity. The so-called "irrational number" is deficient in the criteria of definiteness. The idea of an indefinite number is incoherent, irrational and contradictory. How would an indefinite number work, it might be 2, 3; 4, or something around there? We can speak of an indefinite quantity as a quantity to be measured, when we assume that the thing is measurable, but to say that a quantity has a number is to say that it has been measured and is no longer indefinite. To say that a quantity has a number, but that the number is indefinite, is contradiction pure and simple.

Let's suppose you're standing on the number line at zero looking East toward the positive integers, and you can see the line. How would you describe the geography of the line? If you started to go toward the East, what is the very first thing you'd encounter? A rational number? A "hole" or a gap? Something else?

A number is a definite unit of measurement, a definite quantity. The so-called "irrational number" is deficient in the criteria of definiteness. The idea of an indefinite number is incoherent, irrational and contradictory. How would an indefinite number work, it might be 2, 3; 4, or something around there? — Metaphysician Undercover

In the example of sqrt(2), Alice creates a computer program f(n) for calculating sqrt(2) to n decimal places and sends f(n) to Bob. Bob receives f(n), then he mysteriously decides upon a value for n, then runs f(n) and sends the result to Alice.

I can't relate to your analogy. We haven't established that there is such a thing as a "numberline". I think the other thread demonstrated that there is afundamental inconsistency between numbers, which mark non-dimensional points, and a line which has spatial dimension. The issue is how is the incremental increase (discrete quantity) between two numbers, accurately represented by a line?

My argument is that both of these problems, the incompatibility between points and a line, and the irrational nature of the square root of two, are each a part of the same problem. That problem is the way that we represent space, in dimensions. The spatial representation is incompatible with the numeric representation. The trend in mathematics, at least from Descartes onward, has been to adapt the numerical system to account for the problems encountered by this incompatibility. I am arguing that this is the wrong approach, what is really at fault here, is our spatial representation, and this is what needs to changed.

No, I do not allow that there is such a thing as the square root of two, that's the whole point, — Metaphysician Undercover

Indeed it is. "I do not allow." That says it all.

Do abstractions exist at all? I suggest they don't. A number line "exists" only as an abstraction, but this is not true existence. It's just a concept, in which a set of logical/mathematical properties are considered abstractly. The same is true of numbers, whether rational or irrational. "3" doesn't exist, but collections of 3 objects exist - so we can think abstractly about 3-ness. Neither does Pi exist; nevertheless we can abstractly consider the fact that all "circles" (another abstraction) have Pi as the ratio between their circumference and diameter.

Do abstractions exist at all? — Relativist

As I have pointed out in other recent threads, mathematics is the science of drawing necessary inferences about hypothetical states. Consequently, mathematical existence does not entail metaphysical actuality, only logical possibility in accordance with a specified set of definitions and axioms.

As I have pointed out in other recent threads, mathematics is the science of drawing necessary inferences about hypothetical states. Consequently, mathematical existence does not entail metaphysical actuality, only logical possibility in accordance with a specified set of definitions and axioms. — aletheist

That sounds reasonable. In that vein, do you recognize that there's a conceptual distinction between an "actual infinity" and a "potential infinity"?

In that vein, do you recognize that there's a conceptual distinction between an "actual infinity" and a "potential infinity"? — Relativist

Yes, it corresponds to the difference between metaphysical actuality and logical possibility. Again, mathematical existence refers to the latter, not the former.

cool. Thanks.