at what point in the decimal representation of the surreal does it depart from and differ from sqrt(2), keeping in mind that it cannot be the next largest real number, because that real number is infinitely far away?
I don't know. Is there a way we can construct that?
Reply to tim wood As I understand it, the surreals are not only between all the reals, but also beyond them. They are basically the reals, which are all finite numbers, plus also transfinite numbers, and the reciplocals of transfinite numbers, infinitesimals. Basically, in the surreal numbers, you can do "one over infinity" and get something... not quite exactly zero, but closer to zero than any real number. (But then, it's not exactly just "infinity" that you're dividing by either, but a specific infinite, or transfinite, number).
Also, as I understand it, the surreals are the biggest possible ordered set: every position of every element in every ordered set no matter its size can be represented with a surreal number. There is a particular way of constructing the surreal numbers which is basically equivalent to an infinite process that enumerates every position in an order and every position in between each of those positions and so on and so forth, and in the process of listing out all of those positions, you end up constructing the equivalents of all of the real numbers, but also things that are not equivalent to real numbers, that are in between them and further down the list than any of them. I don't know all the details of that off the top of my head, but I recall it having something to do with building a branching tree of some sort: you start with one element before another, and the other after it, then construct an element before and after each of those, and then before and after each of those, and so on forever. Every node in that tree is a surreal number.
For example, I can say, suppose that between every pair of integers there is an integer. If I'm clever maybe I can work out a system on this "rule." But the idea itself is absurd. The "supposed" integers don't exist. So the question is if surreals have this deficient form of conjectural existence, or do they share the more substantial existence of the reals?
That rule is self-contradictory, so that kind of “number” can’t exist. But you could define a new kind of number — not the integers — which satisfied a property like that. (The rationals, maybe?) Some of that new kind of number would be equivalent to integers, but others wouldn’t.
The surreals are like that relative to the reals. And they exist exactly as much as any other mathematical entity exists, which as you say is a can of worms, but we don’t have to go into it here. Whatever kind of existence mathematical objects may have, there’s no reason to exclude the surreals from it.
The trouble conceptualizing where on the number line they fit is little different from trying to mash the reals in between the rationals. With the rationals you can already find infinitely many numbers between any two numbers, so where do the reals fit? They do though. Same with the surreals fitting in between (and beyond) the reals.
The concept of infinitesimals goes back at least to Leibniz and Newton. Modern day non-standard analysis incorporates these ideas in a legitimate mathematical model. One can prove basic theorems in calculus using infinitesimals, and there have been textbooks that have done that. I once considered teaching an experimental calculus course this way but decided against it. I have heard that for some students calculus is more understandable taught in this non-standard way. I view infinitesimals as metaphysical entities that have achieved a kind of actuality.
The set theory aspects are something else, and seem to appeal to philosophers. Have at it! :cool:
Reply to PfhorrestReply to tim wood The situation with surreals vs. reals is a little different than that with reals vs. rationals. Though it may seem as if rationals completely fill their number line, being as you can fit arbitrarily many points between any two points (i.e. they are densely ordered), rationals are actually full of holes, in a sense. You can construct a sequence of rational numbers that definitely converges to... something. But that something is not a rational number. There are lots and lots of such holes between rationals - in a way, the rational number line almost entirely consists of holes. And that is where real numbers come into the picture: they fill those holes.
With real numbers the situation is different: they are complete. Any converging sequence of real numbers most definitely converges to a real number. There are no holes to fill - at least not in that sense. You have to work a little harder to find what those smug bastards are lacking: you have to violate the Archimedean property.
Surreal numbers are on the number line, unlike complex numbers, which are not. That is, surreal numbers are not complex numbers
Every real number is a complex number, but not vice-versa. There are certain types of surreal numbers that are complex: s = a+bi , where a and b are infinitesimals:
Wiki: A surcomplex number is a number of the form a+bi, where a and b are surreal numbers and i is the square root of ?1.[9][10] The surcomplex numbers form an algebraically closed field (except for being a proper class), isomorphic to the algebraic closure of the field generated by extending the rational numbers by a proper class of algebraically independent transcendental elements. Up to field isomorphism, this fact characterizes the field of surcomplex numbers within any fixed set theory.
You guys are going down a rabbit hole here. :nerd:
The Dirac Delta function (0 everywhere except at x=0, there infinite) can be thought of in terms of infinitesimals
It's funny how ubiquitous the delta function still is in physical and engineering mathematics, and yet it is completely non-kosher from the point of view of standard analysis. It was so useful that it survived Weierstrass's reforms, which did away with non-rigorous infinitesimals of the early calculus.
It's funny how ubiquitous the delta function still is in physical and engineering mathematics, and yet it is completely non-kosher from the point of view of standard analysis
One way to make it kosher is to consider it a generalized function. I never worked with those either.
On the finite number line, infinitesimals are everywhere and nowhere. If r is an infinitesimal then 2+r lies to the right of 2 but to the left of any real number greater than 2. Your job, Tim, should you accept the assignment, is to find it and neutralize it! If you fail we will disavow any knowledge of it. :worry:
The finite number line is a fiction. It may be useful for some things, but to insist that it is somehow 'real' and try to make meaningful inferences from that is meaningless.
The finite number line is a fiction. It may be useful for some things, but to insist that it is somehow 'real' and try to make meaningful inferences from that is meaningless.
I'll pass that on to my colleagues. What a bitter disappointment. :sad:
The finite number line is a fiction. It may be useful for some things, but to insist that it is somehow 'real' and try to make meaningful inferences from that is meaningless. — A Seagull
I'll pass that on to my colleagues. What a bitter disappointment. :sad:
The truth doesn't have to be disappointing. For many it is exhilarating.
Surreals by any other name are just an infinitesimals?
No. I'm old enough that I knew Leibniz, and am familiar only with his basic ideas about infinitesimals. Surreals by Conway are much more elaborate. One can teach a calculus course using only the basic elements of infinitesimals. IMO the surreals are for set theorists and philosophers. Purer mathematics than anything I've done. The real and complex numbers are challenging enough for me!
Reply to tim wood Well, "the number line" in its usual sense is just a visual metaphor for the real numbers (it will do for the rationals as well, though see above about "holes"). So in that sense, no, the number line is not missing anything. You have to work harder to motivate things like infinitesimals and hyperreals. And then you have to work even harder just to reproduce all the things that we can already do with real numbers, like addition and multiplication.
Reply to tim wood Seems to me, in my novice terms, that the left part describes some sequence with the surreal number as its lower limit, and the right number describes some sequence with the surreal number as its upper limit, and that further this mode of representation is powerful enough to produce all the other sorts of numbers... except imaginary numbers, but presumably they could be incorporated with a bit of fiddling. A PhD for someone...
Because they use limits the surreals will presumably be unavailable to those who do not agree that 0.999...=1.
all the other sorts of numbers... except imaginary numbers, but presumably they could be incorporated with a bit of fiddling. A PhD for someone...
Those would be the surcomplex numbers discussed above, numbers of the form (a + bi) where a and b are surreal numbers.
And as there are extensions of the complex numbers into more than just two dimensions, hypercomplex numbers including most notably four-dimensional quaternions and eight-dimensional octonions (beyond which they lose most of the properties that make numbers useful as numbers), you can have surhypercomplex numbers like the suroctonions that I mentioned earlier, numbers of the form (a + bi + cj + dk + el + fm + gn + ho), where a through h are surreal numbers and i through o are the imaginary number and the next seven hypercomplex units beyond it.
(And tetration is the operation after exponentiation, the fourth item in the series {addition, multiplication, exponentiation ...}, hence my joke about "I need to get my suroctonions tetrated").
And as there are extensions of the complex numbers into more than just two dimensions, hypercomplex numbers including most notably four-dimensional quaternions and eight-dimensional octonions (beyond which they lose most of the properties that make numbers useful as numbers),
There's another generalization of the process called Clifford algebras.
I also wanted to mention in passing that the question has been raised as to "where you fit" the extra numbers when you go from the reals to the hyperreals to the surreals and so forth. I believe that the answer is that the claim that "the line in my mind" is the same as any particular mathematical version of a line, is a belief and not a fact that could ever be proven. Is Euclid's line the same thing as the set of real numbers? We take as an unspoken axiom that it is; but if we remember that this is just an assumption, we can resolve our confusion over where the extra points go.
In fact there are a lot of real lines. The constructive mathematicians, who don't believe in noncomputable real numbers, have their constructive line. There are various flavors of the intuitionistic line. The hyperreals form the hyperreal line. It's clearly not the same as the real line, since there are no infinitesimals on the real line.
The surreal numbers are a totally ordered proper class; and if they're totally ordered, we can imagine lining them up in order and calling that the surreal line. But it's not the same line as the standard real line or any of the other many alternative models of the real line. It's a bit of a category error to ask where the extra points go. It's a completely different model of the continuum. That's my understanding, anyway.
Metaphysician UndercoverMay 26, 2020 at 11:00#4162060 likes
I believe that the answer is that the claim that "the line in my mind" is the same as any particular mathematical version of a line, is a belief and not a fact that could ever be proven. Is Euclid's line the same thing as the set of real numbers? We take as an unspoken axiom that it is; but if we remember that this is just an assumption, we can resolve our confusion over where the extra points go.
This is the key point. It is a mistake to try and make the numbers, which represent no specific spatial properties, into a line, which represents a spatial dimension. This is the incompatibility between the non-dimensionality of the numbers, and the dimensionality of the line, which is very similar to the incompatibility between the one-dimensional straight line, and the two-dimensional curved line. They are incommensurate.
What this indicates is that it is a mistake to try and represent spatial existence with distinct dimensions.
The surreal numbers are a totally ordered proper class; and if they're totally ordered, we can imagine lining them up in order and calling that the surreal line. But it's not the same line as the standard real line or any of the other many alternative models of the real line. It's a bit of a category error to ask where the extra points go. It's a completely different model of the continuum. That's my understanding, anyway.
I believe that representing numeration as any sort of line is a fundamental problem. The problem is that a line is meant to represent something continuous, and numbers represent discrete units. The smallest possible spatial unit must occupy all three dimensions of space. So reducing space in this way, attempting to make the smallest possible discrete unit represented by a number, something with one dimension, is a hopeless enterprise because "one dimension" doesn't represent any sort of reality. Therefore an "infinitesimal" must be an infinitesimal point, having position in all three dimensions of space, rather than just a spot on a one dimensional line.
Is Euclid's line the same thing as the set of real numbers? We take as an unspoken axiom that it is; but if we remember that this is just an assumption, we can resolve our confusion over where the extra points go.
@aletheist will be along shortly, I am sure, invoking the ghost of Charles Sanders Peirce and insisting that Euclid's line is not a collection of points at all. He would have a point, at least to the extent that it isn't a given that a line is identical with a particular collection of points.
Reply to SophistiCat
Thanks for the shout-out! I am actually fine with the quoted statement by @fishfry; I have consistently acknowledged that the real numbers are an adequate (though approximate) model of a continuous line for many (perhaps most) mathematical and practical purposes.
?fishfry
Whatever line the surreal is on, I'm still wondering how you would describe writing down the decimal expansion of it.
An infinitesimal is technically a "mathematical quantity", but not a real number. Real numbers have decimal expansions. Infinitesimals have an arithmetic that is not the same as real numbers. r+r=r, e.g. How would that work with a hypothetical decimal expansion? Not all arithmetic is with real numbers.
I tend to look at physics to see what kinds of math are consistent with the physical world. Virtually all I see there is real and complex analysis, functional analysis, group theory, matrix theory, etc. None of which seem related to anything but real or complex numbers. (Well, the Hahn-Banach theorem in FA generates lots of functionals and requires in its proof a single transfinite step, unless this is avoided by requiring a tad more in the hypotheses) . Even string theory - seen as a flop by many - doesn't invoke esoteric number systems.
But I read that surreals are connected to game theory. So what do I know? :roll:
I'm still wondering how you would describe writing down the decimal expansion of it.
There's a decimal-like notation for the hyperreals, called the Lightstone notation.
https://en.wikipedia.org/wiki/A._H._Lightstone
The Wiki article on the Surreals doesn't mention anything about notation, so it's likely that there is no notation yet discovered for them. The article mentioned that the Surreals contain all the ordinals; and there aren't workable notations for all the ordinals as far as I know. I'm under the impression that there aren't even notations for all the countable ordinals but I'm fuzzy on this. Clues might be found here:
There's a decimal-like notation for the hyperreals, called the Lightstone notation
I keep learning things on this forum. I wonder if hyperreals will ever supplant the real number system. Infinitesimal calculus, touted as a more intuitive way to teach the subject, shows up here and there around the world, even in some high schools. The following seems to be a kind of advanced calculus course centered on the hyperreals that complements a similar elementary calculus course that apparently was abandoned years ago:
Reply to jgill Those diagrams make me think of a way of visualizing transfinite numbers I’d thought of before, which I realize now could also be used to visualize infinite’s number.
The transfinite visualization is to imagine the real number line projected sort of logarithmically, so that on the left side you have zero and one the normal distance away from each other, but then the numbers get closer and closer the further right you go until at some finite distance right they “reach infinity”; then you put omega there, omega plus one a single unit right of that, and then repeat the whole logarithmic acceleration until twice omega is twice as far right as omega, then repeat that again. Possibly take that whole new transfinite number line and project it logarithmically the same way to reach even bigger transfinite numbers even faster.
For the infinitesimals, do the same thing, except each “omega” is instead a real number, and the logarithmically projected numbers that asymptotically approach each real number are infinitesimals.
Reply to Pfhorrest Could be. I'm awaiting an application of the hyperreals that is useful in describing or predicting physical phenomena. Perhaps quantum theory will be couched in those terms at some point. But for now they seem to be pretty darn abstract.
Reply to tim wood Thanks for bringing to my attention. But I must admit, going on 89, that having turned off the sound I fell asleep part way through. It seems like it's a discussion about the origins of calculus where the limit definition is given in terms of infinitesimals, called monads or whatever by Newton and Leibnitz. Weierstrass and Cauchy improved upon it by introducing epsilons and deltas.
Non-standard analysis is the modern version, but is not a popular approach in universities.
Comments (42)
I don't know. Is there a way we can construct that?
Also, as I understand it, the surreals are the biggest possible ordered set: every position of every element in every ordered set no matter its size can be represented with a surreal number. There is a particular way of constructing the surreal numbers which is basically equivalent to an infinite process that enumerates every position in an order and every position in between each of those positions and so on and so forth, and in the process of listing out all of those positions, you end up constructing the equivalents of all of the real numbers, but also things that are not equivalent to real numbers, that are in between them and further down the list than any of them. I don't know all the details of that off the top of my head, but I recall it having something to do with building a branching tree of some sort: you start with one element before another, and the other after it, then construct an element before and after each of those, and then before and after each of those, and so on forever. Every node in that tree is a surreal number.
But aren't the surreals a proper class rather than a set?
That rule is self-contradictory, so that kind of “number” can’t exist. But you could define a new kind of number — not the integers — which satisfied a property like that. (The rationals, maybe?) Some of that new kind of number would be equivalent to integers, but others wouldn’t.
The surreals are like that relative to the reals. And they exist exactly as much as any other mathematical entity exists, which as you say is a can of worms, but we don’t have to go into it here. Whatever kind of existence mathematical objects may have, there’s no reason to exclude the surreals from it.
The trouble conceptualizing where on the number line they fit is little different from trying to mash the reals in between the rationals. With the rationals you can already find infinitely many numbers between any two numbers, so where do the reals fit? They do though. Same with the surreals fitting in between (and beyond) the reals.
The set theory aspects are something else, and seem to appeal to philosophers. Have at it! :cool:
With real numbers the situation is different: they are complete. Any converging sequence of real numbers most definitely converges to a real number. There are no holes to fill - at least not in that sense. You have to work a little harder to find what those smug bastards are lacking: you have to violate the Archimedean property.
Every real number is a complex number, but not vice-versa. There are certain types of surreal numbers that are complex: s = a+bi , where a and b are infinitesimals:
Wiki: A surcomplex number is a number of the form a+bi, where a and b are surreal numbers and i is the square root of ?1.[9][10] The surcomplex numbers form an algebraically closed field (except for being a proper class), isomorphic to the algebraic closure of the field generated by extending the rational numbers by a proper class of algebraically independent transcendental elements. Up to field isomorphism, this fact characterizes the field of surcomplex numbers within any fixed set theory.
You guys are going down a rabbit hole here. :nerd:
Is it not more accurate to say that some surcomplex numbers are surreals, or that there is a complex extension of the surreals?
Speaking of which, I need to get my suroctonions tetrated soon...
Picky, picky, picky! :smile:
In over fifty years of complex analysis mathematics I don't think I ever really thought of these critters.
The Dirac Delta function (0 everywhere except at x=0, there infinite) can be thought of in terms of infinitesimals, here in terms of alpha:
[math]\int F(x)\delta_\alpha(x) = F(0)[/math]
It's funny how ubiquitous the delta function still is in physical and engineering mathematics, and yet it is completely non-kosher from the point of view of standard analysis. It was so useful that it survived Weierstrass's reforms, which did away with non-rigorous infinitesimals of the early calculus.
One way to make it kosher is to consider it a generalized function. I never worked with those either.
https://en.wikipedia.org/wiki/Generalized_function
Quoting tim wood
On the finite number line, infinitesimals are everywhere and nowhere. If r is an infinitesimal then 2+r lies to the right of 2 but to the left of any real number greater than 2. Your job, Tim, should you accept the assignment, is to find it and neutralize it! If you fail we will disavow any knowledge of it. :worry:
The finite number line is a fiction. It may be useful for some things, but to insist that it is somehow 'real' and try to make meaningful inferences from that is meaningless.
I'll pass that on to my colleagues. What a bitter disappointment. :sad:
The truth doesn't have to be disappointing. For many it is exhilarating.
That's pretty cool. .
No. I'm old enough that I knew Leibniz, and am familiar only with his basic ideas about infinitesimals. Surreals by Conway are much more elaborate. One can teach a calculus course using only the basic elements of infinitesimals. IMO the surreals are for set theorists and philosophers. Purer mathematics than anything I've done. The real and complex numbers are challenging enough for me!
Quoting jgill
Sure, I shouldn't be surprised that these nasties have long since been tamed, just like infinitesimals and infinities were earlier.
Because they use limits the surreals will presumably be unavailable to those who do not agree that 0.999...=1.
Those would be the surcomplex numbers discussed above, numbers of the form (a + bi) where a and b are surreal numbers.
And as there are extensions of the complex numbers into more than just two dimensions, hypercomplex numbers including most notably four-dimensional quaternions and eight-dimensional octonions (beyond which they lose most of the properties that make numbers useful as numbers), you can have surhypercomplex numbers like the suroctonions that I mentioned earlier, numbers of the form (a + bi + cj + dk + el + fm + gn + ho), where a through h are surreal numbers and i through o are the imaginary number and the next seven hypercomplex units beyond it.
(And tetration is the operation after exponentiation, the fourth item in the series {addition, multiplication, exponentiation ...}, hence my joke about "I need to get my suroctonions tetrated").
This process can be continued indefinitely as in the Cayley-Dickson construction.
There's another generalization of the process called Clifford algebras.
I also wanted to mention in passing that the question has been raised as to "where you fit" the extra numbers when you go from the reals to the hyperreals to the surreals and so forth. I believe that the answer is that the claim that "the line in my mind" is the same as any particular mathematical version of a line, is a belief and not a fact that could ever be proven. Is Euclid's line the same thing as the set of real numbers? We take as an unspoken axiom that it is; but if we remember that this is just an assumption, we can resolve our confusion over where the extra points go.
In fact there are a lot of real lines. The constructive mathematicians, who don't believe in noncomputable real numbers, have their constructive line. There are various flavors of the intuitionistic line. The hyperreals form the hyperreal line. It's clearly not the same as the real line, since there are no infinitesimals on the real line.
The surreal numbers are a totally ordered proper class; and if they're totally ordered, we can imagine lining them up in order and calling that the surreal line. But it's not the same line as the standard real line or any of the other many alternative models of the real line. It's a bit of a category error to ask where the extra points go. It's a completely different model of the continuum. That's my understanding, anyway.
This is the key point. It is a mistake to try and make the numbers, which represent no specific spatial properties, into a line, which represents a spatial dimension. This is the incompatibility between the non-dimensionality of the numbers, and the dimensionality of the line, which is very similar to the incompatibility between the one-dimensional straight line, and the two-dimensional curved line. They are incommensurate.
What this indicates is that it is a mistake to try and represent spatial existence with distinct dimensions.
Quoting fishfry
I believe that representing numeration as any sort of line is a fundamental problem. The problem is that a line is meant to represent something continuous, and numbers represent discrete units. The smallest possible spatial unit must occupy all three dimensions of space. So reducing space in this way, attempting to make the smallest possible discrete unit represented by a number, something with one dimension, is a hopeless enterprise because "one dimension" doesn't represent any sort of reality. Therefore an "infinitesimal" must be an infinitesimal point, having position in all three dimensions of space, rather than just a spot on a one dimensional line.
@aletheist will be along shortly, I am sure, invoking the ghost of Charles Sanders Peirce and insisting that Euclid's line is not a collection of points at all. He would have a point, at least to the extent that it isn't a given that a line is identical with a particular collection of points.
Thanks for the shout-out! I am actually fine with the quoted statement by @fishfry; I have consistently acknowledged that the real numbers are an adequate (though approximate) model of a continuous line for many (perhaps most) mathematical and practical purposes.
Ha! Saw what you did there...
My understanding is, none.
Perhaps until someone builds some more numbers.
As I understand it, it is somehow provable that there cannot be more numbers than the surreals.
Yes, that's what I understand, too.
But one ought not underestimate the power of human creativity...
An infinitesimal is technically a "mathematical quantity", but not a real number. Real numbers have decimal expansions. Infinitesimals have an arithmetic that is not the same as real numbers. r+r=r, e.g. How would that work with a hypothetical decimal expansion? Not all arithmetic is with real numbers.
I tend to look at physics to see what kinds of math are consistent with the physical world. Virtually all I see there is real and complex analysis, functional analysis, group theory, matrix theory, etc. None of which seem related to anything but real or complex numbers. (Well, the Hahn-Banach theorem in FA generates lots of functionals and requires in its proof a single transfinite step, unless this is avoided by requiring a tad more in the hypotheses) . Even string theory - seen as a flop by many - doesn't invoke esoteric number systems.
But I read that surreals are connected to game theory. So what do I know? :roll:
Quoting tim wood
There's a decimal-like notation for the hyperreals, called the Lightstone notation.
https://en.wikipedia.org/wiki/A._H._Lightstone
The Wiki article on the Surreals doesn't mention anything about notation, so it's likely that there is no notation yet discovered for them. The article mentioned that the Surreals contain all the ordinals; and there aren't workable notations for all the ordinals as far as I know. I'm under the impression that there aren't even notations for all the countable ordinals but I'm fuzzy on this. Clues might be found here:
https://en.wikipedia.org/wiki/Ordinal_notation
I keep learning things on this forum. I wonder if hyperreals will ever supplant the real number system. Infinitesimal calculus, touted as a more intuitive way to teach the subject, shows up here and there around the world, even in some high schools. The following seems to be a kind of advanced calculus course centered on the hyperreals that complements a similar elementary calculus course that apparently was abandoned years ago:
https://www.math.wisc.edu/~keisler/foundations.pdf
The author even has diagrams showing "where" infinitesimals and transfinites are located on the real line! :cool:
The transfinite visualization is to imagine the real number line projected sort of logarithmically, so that on the left side you have zero and one the normal distance away from each other, but then the numbers get closer and closer the further right you go until at some finite distance right they “reach infinity”; then you put omega there, omega plus one a single unit right of that, and then repeat the whole logarithmic acceleration until twice omega is twice as far right as omega, then repeat that again. Possibly take that whole new transfinite number line and project it logarithmically the same way to reach even bigger transfinite numbers even faster.
For the infinitesimals, do the same thing, except each “omega” is instead a real number, and the logarithmically projected numbers that asymptotically approach each real number are infinitesimals.
Non-standard analysis is the modern version, but is not a popular approach in universities.