In a recent OP I looked at a discovery by mathematician Norman Wildberger, who found a general method for generating power series solutions of polynomial equations of any degree. Wildberger has an interesting, extremely unconventional and (in my opinion) flawed philosophy of mathematics, which among other things denies the existence of irrational numbers. Here he explains why √2 doesn’t exist, at least not in the way that mainstream mathematicians thinks it does:
There’s lots to criticize about this, but I’ll save it for the comments.
Keith wrote to jock, “What is the ‘it’ that you square in order to get 2, if √2 is not a number?”
I don’t see that Jock’s reply addressed that question.
aleta,
Did you watch the video? Specifically, the algebraic approach to root2?
aleta:
It did, indirectly. Jock is referring to Wildberger’s “algebraic √2”, which isn’t a number. It’s an algebraic symbol with the property that when you square it, you get 2. It’s analogous to how when you square i, which isn’t a real number, you get -1, which is a real number.
The problem is that Jock is trying to use it to represent a length. It can’t do that, since it isn’t a number. More on that in a subsequent comment.
Jock,
Here’s Wildberger at 27:48 in the video, referring to his “algebraic √2”:
I didn’t watch the video. I guess I see that “algebraic root 2” isn’t a number, but something else?
I wonder what sine 31 is, or e^2), or ln 7? Are these algebraic identities, with various properties, but also not numbers.
I know keith has been explaining this, and I don’t think it’s a viable, useful, or even defensible distinction.
aleta:
It’s a (defective) way for Wildberger to have his cake and eat it too. He doesn’t want √2 to be a number, but he does want to have an object that can (sort of) play the role of √2. So he defines this abstract object — the algebraic √2 — that has the property that (√2)^2 = 2, but isn’t a number and doesn’t have a decimal expansion. More on this later, but my point for now is that the algebraic √2 isn’t a number, so Jock can’t use it to represent a length.
aleta,
The segment of the video that describes the algebraic √2 is only about 5 minutes long, so if you want to watch it, it starts here.
🙂 Videos are a poor way of conveying information. Some can be useful. I rarely get my toolbox out these days without searching “how to”.
Computers allow visualizations of concepts that I found difficult to grasp when the tools were textbooks, biro, blackboards and an eccentric headmaster. I do recall being fascinated by the connection between trigonometry and algebra when we got on to de Moivre’s theorem (formula, these days). Not until much later did I learn it is a short step to Euler’s identity.
And substituting
for 
Never found it useful since, though. Measurement, calculation, arithmetic, mathematics. Everything is connected
ETA must stop playing with
Jock:
The things on it are points, and each point is assigned a value — a number. The points are connected, with no gaps, because connectedness is a property of lines. Lines are infinite in length, and they can be divided into line segments that have finite lengths. Lengths are very much a feature of the number line, as they are of any line. In fact, the number assigned to a given point on the number line tells you the length of the line segment connecting it to the origin — the point that has the number 0 assigned to it.
Length can vary continuously. Start with the line segment connecting the point 0 to the point 1. Now extend the line segment smoothly to the right, stopping when the right endpoint reaches 3. As the line segment extends, the right endpoint sweeps through all the values between 1 and 3, both rational and irrational. At one point it is √2 units away from the origin, and at another point the distance from the origin is 2. If length can vary continuously, why shouldn’t the numbers that are used to express length also vary continuously? Why not label the first point “√2” and the second point “2”?
You’ve acknowledged that √2 has a value:
…so why deny it the status of number? The fact that you “can’t get there” via a finite decimal expansion doesn’t mean that the value doesn’t exist. You’ve confirmed that it does exist, so why not do as we do and call the value a number? Why put yourself in the odd position of claiming that some values are numbers and others are not?
And yet have zero volume. (Imaginary ones at least)
Alan:
Yes. Also zero area and zero width. They’re ghostly little fuckers. So to speak.
Quick question: does an omniscient God know the exact value of √2?
That is because they are mere concepts in Mindspace.
faded_Glory:
Definitely. Piece of cake for an omniscient God. Wildberger actually mentions a hypothetical “God’s garden” in which there is “a beautiful, infinite stone” on which all the digits of √2 are written.
Which is actually relevant to our discussion, believe it or not. That’s because if God knows all the digits, then √2 is a completed object by Wildberger’s standards and thus eligible for numberhood.
You just keep doing this. It’s pretty entertaining.
For a moment I thought you had found a much, much better argument (well, any argument at all…) for your position with your “one point it is √2 units away from the origin, and at another point the distance from the origin is 2” — I thought “wait a sec: if we consider the number line of ln(2), then √2 is at the midpoint between 1 and 2 and maybe I can argue that this midpoint must exist.” Then I realized – if logbasex2 is rational, then so is logbasex√2 , and if it is irrational, so is the other. We’re back with the algebraic approach. Oy veh.
Finally, an argument.
FG says, “Does an omniscient God know the exact value of √2?”
Keith responds, “Piece of cake for an omniscient God. Wildberger actually mentions a hypothetical “God’s garden” in which there is “a beautiful, infinite stone” on which all the digits of √2 are written.”
How ridiculous to think that God’s knowledge of numbers is in our decimal representation system! 🙂
Does God know all the integers is a more relevant question in pondering whether God can comprehend a completed or actual infinity. But it’s an irrational, imaginary question, as the idea of God is just an created abstraction about which our thoughts are just a projection of some fictional assumptions.
So this really doesn’t further the discussion, I think.
aleta:
Just to be clear, in case there’s any doubt, Wildberger isn’t trying to make a theological point here. I don’t even know that he is a theist. His point is that it’s hubristic of us, as mere humans, to think that “we can understand this number” (√2 ). I guess in his view we have to know all the digits in order to “understand” a number. (I like the fact that he slipped and called √2 a number, though.)
Of course, an omniscient God would know all the digits of √2 in any base. Would he choose base 10 for his infinite stone? Some Christians might argue that he would, because we were created in his image and he therefore has ten fingers.
(Yes, some Christians take “in his image” quite literally. They don’t necessarily mean that God has a physical body (though Mormons do), but just that his spiritual form “looks” like us, complete with spiritual head, arms, hands, toes, etc. And presumably spiritual teeth and a spiritual gall bladder.)
I’m aware that this isn’t really a theological topic. But is seems that we can’t help but think about whether it would be possible to comprehend an actual infinity all at once. I’m going to think about that more.
Everything is connected. 🤓
Jock,
There’s a lot to untangle here. You wrote:
You’re missing something fundamental about the number line, which is that the numbers are evenly spaced. The distance between 2 and 3 is the same as the distance between 338,767,206 and 338,767,207.
You seem to be trying to define a new line in which the value assigned to a point is equal to the natural log of its distance from the origin. That’s not the number line; it’s a transformation of the number line in which x’ = ln(x).
Second, I don’t know why you referred to it as “the number line of ln(2)”. ln(2) just specifies a point.
Third, ln(x) is undefined for 0 and negative numbers. You don’t have much of a number line if it only covers the interval (0, ∞].
Fourth, there’s really just one number line. Every point on the line is assigned a number, not a representation of a number. There’s a point on the number line that corresponds to the abstract number 5, but I can label that point with any of the representations of 5, including “5”, “V”, “101” (binary), “10/2”, and so on. Numbers are distinct from their representations, and the distinction is crucial, which is why I jokingly suggested that you and Flint tattoo a reminder of that on your forearms during our epic 8-month discussion.
So my point stands:
Jock,
I’m also interested in your answer to the question I posed earlier. You’ve affirmed that √2 has a value, so why deny that it is a number? Why put yourself in the position of claiming that some values are numbers and others aren’t?
Is existence discrete or continuous?
aleta:
I suppose it depends on what you mean by “comprehend”. I would say that I comprehend the infinity of the natural numbers, but all I mean by that is that I can understand a set in which there is no greatest number. I comprehend the fact that every element in that set has a successor and that no two numbers share the same successor, and a set in which that holds is necessarily an infinite set. In that sense you could say that I comprehend an actual infinity.
If comprehending that set means holding all of the natural numbers in my mind at the same time, then of course I can’t comprehend it. I can’t even hold all the numbers from 0 to 20 in my mind simultaneously, much less the entire set. God, with his infinitely capacious mind, could pull it off.
‘Define “exist”‘ seems to be the challenge nobody here is prepared to take on.
No idea. Can you explain what that has got to do with my question?
There is no such thing as ‘all the natural numbers’. We can define a concept called the *set* of all the natural numbers but we can’t define all the numbers in that set, because whatever you name as the last one there is always another one after that. I’m not at all convinced that an omniscient God could keep them all in his mind, because there is not really an ‘all’.
The set is an interesting concept that we can play with and use, but trying to grasp all its members is futile, just as it is futile to try and define the exact decimal value of √2.
This, at least, is correct. I was trying to find a version of your “argument” that might have a chance of working. Unfortunately it lands right back where we started: sorry I couldn’t help you.
i think not 😉
Seriously, though, when you write stuff like this:
Do you understand why I find this so immensely entertaining?
Well, for starters, I have never denied that √2 is a number. Your ability to repeatedly mischaracterize what others have written is amazing.
√2 has a value, but we can never get to it. Therefore there is a fundamental problem with “placing it on the number line” with the rationals.
It is a flintjock number.
I’ll get me coat.
I think the key difference is again between a potential and actual infinity. FG writes, “We can define a concept called the *set* of all the natural numbers but we can’t define all the numbers in that set.” I agree with this sentence, but I think the second “define” in “define all the numbers …” is the wrong word to use. We can’t list (enumerate), or even imagine listing, all the numbers because there are always more. What we can do is what Cantor did: define a new abstract mathematical concept N, the set of all whole numbers, and then learn to deal with that concept logically in relationship to other concepts. N represents an actual infinity. The concept N exists just as other mathematical concepts exists, but all the numbers themselves don’t exist, even mathematically, as they are part of a potential infinity.
keith reminded us of how we used to argue with Kairosfocus that “all numbers are finite, but there are an infinite number of them.”
So what does it mean to “comprehend” all this? I think it means to abstractly understand both these things about infinity: that the set of natural numbers has no limit–there is always one more–and that the concept of the set of all natural numbers can be said to exist within the field of set theory. All these things exist as abstractions in human minds, manifested as verbal and written symbols within logically coherent mathematical systems.
aleta,
Given the theological context, it did cross my mind that you might have been using “comprehend” in its John 1:5 sense — surround or (on point here) “grasp in its entirety”.
Jock, that would be the theological way to understand the word, but human comprehension is different. And as I somewhat flippantly said earlier, talking about the comprehension of God is an exercise in making things up about which I have no actual belief, so I don’t think it’s very relevant.
aleta,
I agree.
faded_Glory:
Then you might actually be a finitist!
Yes, but notice that you’re defining it as including “all the natural numbers.” How can it include all the natural numbers if there is no such thing as “all the natural numbers”, as you said in your first sentence?
You can’t name the last one because there is no last one. That’s what makes the set infinite.
And as aleta noted, “define” really isn’t the right word here. You really mean that we can’t list all the numbers, right? That’s true, but it doesn’t mean that we can’t define them. We can define them as follows:
1) 0 is an element of the set.
2) for every n in the set, n + 1 is also in the set.
That doesn’t list the elements, but it does define them.
That depends on what you mean by ‘grasp’. It’s the same issue as with the word ‘comprehend’, which I described in my previous comment.
Be careful. Technically, there’s no such thing as “the exact decimal value of √2”. It has an exact value, but that value isn’t a “decimal value” because numbers are distinct from their representations. Values can be represented in a number of ways — decimal, binary, octal, Roman numerals, etc — but a value is just a value, regardless.
Can I define the exact
decimalvalue of √2? Yes. It’s the number x such that x^2 = 2. That defines it without listing all of its decimal digits.aleta,
I agree with everything you wrote here except for the following:
Numbers are mathematical concepts, too, just like ℕ, so if you regard ℕ as existing, why not the numbers within ℕ? In fact, how can ℕ be said to exist if some of its elements are missing?
You also run into the kinds of problems I mentioned earlier in the thread. If the natural numbers don’t all exist, but they’re potentially infinite, who is creating them, and at what rate? Which ones don’t exist yet? Why should time be a factor? Math isn’t constrained by the limits of time and physics. The successor function S(n) = n + 1 should in my opinion be regarded as operating instantaneously, or better still as operating outside of time altogether. If you treat it that way, then the set ℕ really is complete, because it took literally no time at all for all of its elements to be generated.
I agree with your clarification, keith. All the whole numbers exist, which is a separate issue from the process or arriving at them by stepping through the process of listing them by following the definition of a successor. 1774^674 exists, which is different than the representational problem of writing it out in decimal format, or arriving some how at (1174^674) – 1 in order to bring 1774^674 into existence.
By the way, this business about having to go through all the steps to “create” the whole numbers was, I think, one of the places where Kairosfocus got hung up.
keiths:
Jock:
If you agree, then what was the point of introducing a “number line of ln(2)” that isn’t a number line, isn’t based on ln(2), doesn’t include the interval [-∞, 0], and doesn’t advance the discussion?
Thanks for trying to help, but I’m not seeing the need. What specifically is wrong with the argument I’ve presented?
keiths:
Jock:
I swear it’s true. Try it on your calculator, or follow this link. Your “number line” is missing the infinitely wide interval [-∞, 0]. That isn’t very number-liney.
No.
keiths:
Jock:
You evicted it from your number line. Why did you evict if If you don’t deny that it’s a number?
You don’t have to place it on the number line, nor do you have to place the rationals there. All of the numbers, both rational and irrational, are already there. The number line is complete, with no missing values. That’s why it’s called “the continuum”.
You yourself confirmed that √2 is already there, writing:
If you can’t place it on the line, as you say, but you can define an interval that contains it, then it’s already there.
It can’t be. You noted that it has a value, singular, which means it cannot be a flintjock “number”, since those are multi-valued.
(For the benefit of onlookers, the flintjock numbers that Jock is referring to are “numbers” (that aren’t numbers) that he and Flint invented for use in expressing measurements. They were motivated by the mistaken belief that real numbers, which are exact (ie infinitely precise) cannot be used to express measurements, which are inexact. It was one of the two main topics of our epic 8-month discussion, the other one being whether “3 = 3.0” is a true statement, lol.)
aleta:
Yeah.He was hung up on the fact that a stepwise process of incrementing would never produce a transfinite number. Which is true, but irrelevant, since the infinite set of natural numbers doesn’t contain any transfinite numbers. They’re not there, so you don’t have to reach them.
Ln (negative real number) is a complex number, which obviously doesn’t lie on the real number line. ln(0) is undefined. This is only relevant because for some reason Jock wanted to introduce a number line based somehow on ln x, but I don’t know why.
I know rehashing Kairosfocus is silly, and off-topic, but nostalgic. What I liked was if you write N = 1, 2, 3…., the transfinites were “beyond the ellipsis”. 🙂
aleta:
Yeah, his explanation didn’t make much sense:
Perhaps he’ll tell us what was going on in his head when he wrote that.
I remember he referred to the place “beyond the ellipsis” as some kind of “zone”. The Twilight Zone, maybe?
I accept the various alternatives to the way I formulated my previous post. In my defense, all I can say is that a) English is not my native language, and b) I can often get myself into a real muddle 🙂
Are we drifting off into a definitional argument? Does it depend on how you define the number line whether √2 is on it, or not? If the line is defined as containing all the real numbers then it is on it. If the line only contains the rational numbers, it isn’t. Simples?
Both God and numbers are made up, in my view. Without human minds to create them they wouldn’t exist in the world.
Even Clever Hans needed a human to do his maths 😉
faded_Glory:
It’s definitional to an extent, because there’s no law against choosing your own definitions. Eugene might define “number line” in a way that banishes the primes. Stephanie might leave out all the numbers between 761 and 762. Javier might want to stuff the imaginary numbers in among the reals. Katherine might want the numbers 44 and 2 right next to each other. Jock doesn’t want the irrationals on his number line.
They’re free to make those choices, but that doesn’t mean that their choices are coherent, clean, and useful.
My feeling is that we should stick to the standard definitions unless we have good reason to substitute our own. The standard definitions make communication easier and have usually been well tested. While there sometimes are good reasons to switch definitions, I haven’t seen that in this discussion.
Jock’s number line isn’t a line. He described it as “a series of dots”, and he put the word “line” in quotes n order to emphasize that it isn’t truly a line. He has oddly banished the irrationals from his line, despite the fact that he recognizes them as numbers and despite the fact that there is room for them on the line. He has argued that √2 can’t be placed on the number line, yet it is already there, as he himself has acknowledged. I’m unaware of any reason to think that Jock’s definition of the number line is preferable to the standard definition.
It seems odd to me that people would argue about definitions and simultaneously assert that everyone’s definitions are commensurate and functionally identical.
It would seem that the business of mathematics is to produce definitions that are self consistent, and that this task is Sisyphean.
The square root of 2 can be defined as that number which, multiplied by itself, produces 2, and at the same time, we can say that it is impossible to write out the number exactly.
I can remember, in high school, being upset at being told that division by zero is undefined. I got over it, sort of.
When i asked whether existence is continuous or discrete, I was pondering whether an irrational number is continuous or discrete. Can it be said to occupy a position on a line if it cannot be instantiated? Could we not say its position is undefinable? There are no definable numbers that it fits between. Could we not admit that some kinds of definitions lead to contradiction?
petrushka:
I haven’t seen anyone in this thread taking that attitude. Who are you thinking of?
The definitions of the number line that we’ve been discussing have vastly different functional consequences. For instance, the standard number line is continuous and connected, containing all numbers, while Jock’s number line is, as he describes it, “a series of dots”. It’s what you get when you remove all the irrationals, leaving gaps behind. The “dots” are what remain, each corresponding to a rational number. Those dots occupy 0% of the space on the original number line.
Standard number line: 100%. Jock’s number line: 0%. That’s a pretty big functional difference, and it’s due to a difference in definitions.
It’s impossible to write out the number as a decimal, but it is possible to write down the exact value. Just write “√2”. I’m not being flippant — that is a representation of the exact value of √2, just as “13” is a representation of the exact value of 13, or “24 + 2” is a representation of the exact value of 26.
It’s easy to lose sight of this, but using expressions to represent numbers is something we do all the time. “573” really means “5 × 10² + 7 × 10¹ + 3 × 10⁰ “, but the former is more compact and so we use it for that reason. There is no single symbol for 573, so we have to specify it as a sum, using three digits.To get the exact value, you have to do the addition.* Specifying 26 as “24 + 2” is no different in principle.
Likewise, “√2” specifies the exact value of √2. You just have to run a square root algorithm to generate the decimal digits, and since there are infinitely many of them, you can’t generate them all. You have to stop at some point and accept that the representation you have produced is of a number that is only approximately equal to √2.
Another way to represent the exact value of √2 is to write it as “10” in base √2. In that base it has a nice, finite representation. “10” just means “1 × (√2)¹ + 0 × (√2)⁰ “. Base √2 isn’t particular useful — I’m just pointing out that there is a compact way of representing √2 using digits. Numbers are distinct from their representations. √2 has both finite and infinite representations.
Wildberger is your man. In one of his videos, he defines an extension to rational number arithmetic in which division by zero is allowed and ∞ is a number.
An irrational number is discrete in the sense that it has one and only one value, as do all numbers, but it’s part of the continuum, which (as the name implies) is continuous.
I’ll assume that by “cannot be instantiated” you mean “cannot be written as a finite decimal”. Yes, √2 occupies a point on the number line despite not having a finite decimal expansion.
Earlier in the thread, aleta explained how to locate that point:
petrushka:
Its position is definable as “the point that lies exactly √2 units from the origin, on the positive side”. It can be located using aleta’s method.
There are infinitely many pairs of definable numbers that √2 fits between: -30 and +30, 0 and 5, 1 and 2, 1.4 and 1.5, 1.414213561 and 1.414213563. They can be arbitrarily close together, and √2 will fit between them.
Absolutely. We can, and we do, but do you think the standard definition of the number line leads to some kind of a contradiction?
* Computing the sum will just get you 573 all over again, but that’s only because we default to using decimal representations.
Sqrt 2 can be exactly defined by a procedure or by an operation. This is conceptually different from the way we think of whole numbers or rational numbers. If it were not different, we would not need categories.
petrushka:
So can 573. It’s 500 + 70 + 3, exactly.
√2 falls into a different category than the whole numbers or the rational numbers, but what distinguishes it isn’t that it can be exactly defined by a procedure or operation. It’s that it can’t be expressed as an integer ratio. Hence “irrational”.
Also, there are plenty of irrationals that, unlike √2, can’t be exactly defiined by a procedure or operation. They’re called “uncomputable numbers”.
Time to discuss Wildberger’s “algebraic √2” in more detail.
√2 isn’t a number, according to Wildberger, because it has an infinite decimal expansion. In fact, it isn’t even a legitimate mathematical object, because it hasn’t been “completed” to Wildberger’s satisfaction. Yet he would still like to have a mathematical object that can (sort of) fill in for √2, having the property that when it is squared, the result is 2.
His solution is to define an object he calls “√2” that isn’t a number and doesn’t have a decimal expansion at all, much less an infinite one. He then extends his number system by including this new object and defining its behavior.
Since this new √2 object isn’t a number, it doesn’t occupy a spot on the number line, but rather sits off to the side in the same way as the imaginary number i does.
He combines existing numbers with multiples of this new √2 to create new mathematical objects of the form “a + b√2”, where a and b are both rational numbers. These objects can be manipulated according to the standard rules of arithmetic.
For example, to add two of these objects, you add their corresponding components, just as you would for complex numbers. In other words,
To multiply them, you just do ordinary binomial multiplication:
So far, so good. But there are serious downsides, which I’ll discuss in a later comment.
One serious downside is that you also have sqrt 3 and cube root 17 and sin 31 and ln 7 and (dare I say it) an infinite of cases which would have to be handled this way. Every irrational would need to have an “algebraic” definition. What would be the operational definition of ln 7?
Moreover, if √2 can’t be placed on the number line because its exact value as a decimal expression is unknown, then neither can 1/3 which has an infinite series of 3’s behind the decimal point.