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I'm especially interested in an approach to mathematics that involves treating large numbers as if they are infinite. This is a little unorthodox, but a recognised approach that some mathematicians have followed from time to time.
For instance, 100.......00, with a thousand zeros, is well beyond anything one would reach by addition of the ordinary sorts of numbers people use for money, or measuring things about the house. However one can get to it quickly by multiplication. Start with 2, and keep squaring, and you get beyond it in twelve steps. So let's try for something larger.
Now define n as one followed by a thousand zeros. Then the number 100......00, with n zeros, is well beyond anything one would ever reach even by multiplication, starting from ordinary numbers. One can't even begin to imagine the scales defined by these numbers.
Some specialist areas of mathematics need much larger numbers than that.
Even these can have their limitations. If you define the methods of a particular field precisely, this will limit the size of the largest numbers that can be defined in practice in that field.
This is true even for the specialist area of mathematics which has rapidly growing sequences as it's particular study. If you try to limit the range of methods used, and write them all down precisely, you will limit the largest size of number you can define in practice.
By inspecting the methods of any field of mathematics which is exactly described, one can extend it to another field that can make even larger numbers.
One way to do this is with a reference to the largest numbers that can be defined by the methods of the field in a given number of steps. By referring to the methods of the field in this way, one gets short definitions of numbers well beyond any that can be defined in practice within the field itself.
One could keep going, and make it a project to define the largest numbers one can in practice, with more and more concise definitions. One could imagine whole communities of mathematicians working on the project, and maybe imagine some extremely large number as the largest they would make after so many years of work. They almost certainly wouldn't know which of all their numbers is the largest, but they could define a new number as the maximum of all the ones constructed, and so make it the largest one by definition.
Just imagining such a project doesn't by itself define any new large numbers, but it hints at the possibility of new numbers beyond all the ones we can make at present.
One is talking here all the time about what's possible in practice, to see what happens if one refuses to get dragged into any abstract idea about going on for ever, which one perhaps doesn't completely understand.
My assumption is that there are finite numbers beyond all the ones we can construct, refer to, or hint at. I take it as an axiom that there are finite numbers beyond the horizon of all our practical limits. These are the ones that one can treat as infinite in some sense. If one is willing to restrict ones methods to a particular field, one can hope to treat smaller numbers than these as infinite.
Some areas of mathematics use the idea of infinity itself as an abstract concept. You come across it early on, in calculus. To find the area under a curved shape, mathematicians divide it up into rectangles, or other shapes they know the area of, together with some left over areas of irregular shape. They can plan the sub-divisions in such a way as to limit the total size of the left over areas. By using smaller sub-divisions, you get more accurate results.
For instance, suppose you know that an area is between 1.9 and 2.1, and also between 1.99 and 2.01, and in fact that you can get as close as you like to 2 from either side. Then a mathematician would say that its area is 2, accurate to as many digits as you like.
But in practice, one doesn't need an accuracy of millions of digits. It's enough to know that it is between 1.9999 and 2.0001 perhaps. On another occasion one might want a little more accuracy, but not really that much mathematically speaking. So one might wonder if it is possible to get by without mentioning infinity at all. And indeed one can. This was established in a paper by Mycielski. He showed how to develop calculus without mentioning infinity at all. This is possible provided that one accepts limits on the lengths of proofs.
The approach described here has traditionally been linked with the idea of mathematics as a construction, rather than as something discovered. But it doesn't have to be. Mycielski's paper uses the logic of discovery (classical logic).
Whatever your view, you have to think of some parts of mathematics as constructed.
For instance, the numeral '2' (the character we use) is a construction, and isn't the number two itself. Nor is the number two any particular pair of objects, nor the class of all pairs of objects. Those are all constructions that have been used to help explain or understand it.
If the number 2 is discovered rather than constructed, what it is that is discovered is somewhat elusive. We work with it using many constructions. So with that approach one has to think of constructions as a necessary aid to discovery.
If there can be finite numbers beyond ones horizon, which one can never reach, it makes sense to speak of a finite number with an unending sub-sequence.
Of course, there will be lots of such numbers, but let's start with one of them.
None of our practical constructions of numbers can ever reach it. For us, the sequence of all the numbers we can construct is unending. So for us, the sequence of number leading up to it is unending.
Such a number belongs to the realm of discovery. One mightn't be able to say much about it, except that there must be such numbers beyond one's horizon. The unending sequences by which one studies it are constructions.
I support an intermediate position according to which finite numbers, including the ones beyond ones current horizons, can be discovered, and unending sub-sequences of large finite numbers are constructed.
I call a large finite number "seemingly infinite" if it has an unending sub-sequence.
For instance one can start with small numbers, and keep adding numbers, and never reach a number with a thousand zeroes. That's true if adding by hand, or using computers, including any computers in the foreseeable future. So, if one has no more powerful methods than addition, this number counts as seemingly infinite. The numbers themselves are discovered, while any unending sequences are constructed.
Note that with this approach, there is no smallest seemingly infinite number. You can construct new numbers by starting with small ones and building up, or by starting with large ones beyond the horizon, and reducing them in size. For instance, if you subtract one from a seemingly infinite number, the result is still seemingly infinite. Also if you divide by two, it is still seemingly infinite, and so on.
A number beyond the horizon needs to be given in advance, as a constant of the theory, and from this one can construct more of them.
The two logics of discovery ("Classical logic"), and of construction, are identical for finite numbers, but differ for unending sequences.
If one thinks of a sequence as discovered in its entirety, then any particular type of number has to be either in it or not in it.
For instance, there is a sequence of numbers called the Fermat numbers. The first few members are 3, 5, 17, 257, 65537. These numbers are all prime (have no divisors except 1, and the number itself). All the numbers known from then on are composite (have divisors). Are there any more prime numbers in this sequence? If one uses the logic of discovery, one can say that, though we don't know the answer yet, either there is or there isn't another prime in the sequence.
If one uses the logic of construction, one can only say what one knows, or has a method to find out that is guaranteed to yield an answer one way or another. So far, nobody has found any such method for the Fermat numbers. We have no way of knowing whether there are any more primes in it or not, and no line of investigation has ever been suggested that is likely to decide the issue one way or another.
So one would just leave it open as to whether there is another prime in the sequence. One simply doesn't say anything about it if asked.
One assumption sometimes made is that constructive logic is time dependent. For instance, Fermat's last theorem used to be used as a stock example of a result not yet proved or disproved. But now it's been proved.
So was it always true or did it become true when it was proved?
This is linked with the idea of a truth value of "not known". If it became true when proved, it must have had an indeterminate truth value before it was proved.
According to the logic of discovery, any statement has to have a "truth value" such as true or false, and mathematicians or philosophers used to this logic often assume that the logic of construction has a third truth value of "not known". That idea is even sometimes printed in exegesis of the logic of construction by other researchers outside the field. But actually, using the logic of construction as it is most usually presented, one can prove that there is no third truth value. It's quite a short proof too (click here to see).
Using the logic of construction, a conjecture is true if proved. It is false if shown to lead to a contradiction. One can say that it is either true or false if there is a method which is guaranteed to decide the question in a finite number of steps, even if it hasn't been decided yet. If none of those apply, one can't say anything at all about the truth/falsity.
So once Fermat's Last Theorem is proved, it is proved to have been always true. But before it was proved, and before you had any method guaranteed to decide it in a finite number of steps, you couldn't say anything about whether it is true or false.
One has to get used to the idea of not saying anything at all about something if one can neither prove it nor disprove it nor provide a method to decide whether it is true or false.
If one treats large finite numbers as seemingly infinite, then unending sequences can be of varying sizes, depending on the context of how small a number can be and still be considered as beyond the horizon of the numbers one is treating as seemingly finite. The positions of the horizons of the unending sequences will depend on the power of the methods used in the field to construct finite numbers, and the resources of time, space etc available to define new numbers.
Then it is possible that some things one might say about an unending sequence could be true for some notions of seeming infinity, false for others, or have to be left open without a truth value, depending on how extensive ones notion of seeming infinity is. For instance, with a less extensive notion of seeming infinity, it might be false to say that there are more numbers in a particular sequence, while with a more extensive notion, the sequence might have more numbers yet to be found.
So in principle, one could add in new axioms that only work for less extensive notions of seeming infinity.
Howerver, normally, when setting out the logic, one will make no assumptions about the size of the number beyond the horizon, so that anything one can prove using a more limited notion of seeming infinity will remain true when one adds new methods, and expands one's horizons to a more extensive notion.
If one follows this approach, one can still use the logic of discovery for finite realms of discourse, since the logic of discovery and of construction is identical for these.
In particular, one can use it for proofs of bounded size for clearly stated conjectures in sharply defined theories.
Take the example of the Fermat primes. Two conjectures are, that there is another prime to find, and that there are no more primes in the sequence.
Then there either is or isn't a proof of one of these results in less than a million steps (say) in the standard presentation of the theory of numbers, because the total number of such proofs is finite, and they can be enumerated, and checking whether any particular proof is correctly stated and leads to the desired conclusion is a finite process.
This leaves it open to one to decide whether one thinks of proofs as discovered, or constructed, depending on ones philosophical views.
Here, the idea of seeming infinity introduces something new, because one can consider the possiblity of a finite realm of discourse that extends beyond ones horizon of all the numbers one can construct in practice.
The logic of discovery is identical to the constructive logic for ordinary finite numbers. So since this approach has finite numbers beyond the horizon of unending sequences, one can use the logic of discovery for those.
One only needs use constructive logic for the unending sequences themselves.
This gives a stronger theory, especially for calculus. It is closer to the classical approach than the usual constructive theories.
The most awkward thing about conventional constructive approaches to calculus is that one can't use something called the trichotomy principle. So what is the trichotomy principle?
Many numbers are only given by infinite sequences. For instance, pi is 3.14159... The "..." means that the digits go on for ever. Pi can be defined in various ways using infinite sequences, for instance, you can use the area of a square, of an octagon, and so on, doubling the number of sides each time, and approximating closer and closer to the area of a circle of radius 1.
To define a number in this way, all you need is a method for finding a sequence of closer and closer approximations to it, that can get as close to it as you like. Then if you continue far enough along the sequence, you can find out the number to as much accuracy as you wish.
But now suppose two numbers x and y are both defined using infinite sequences. One now wishes to know if x > y or x < y or x = y. The first two are easy. If x is larger than y, you will find this out if you calculate long enough, and similarly if x is smaller than y. But what if you keep calculating and never find out which is the largest.
According to the logic of discovery, either x > y or x < y or x = y. This is called the trichotomy principle. However you might never be able to prove any of these, and you might have no method guaranteed to decide the question.
Let's take an example, let x = 1/3 + 1/5 + 1/257 + 1/65537. Let y be the sum of the reciprocals of all prime numbers in the sequence of fermat primes. Then according to the logic of discovery, x < y if there is another fermat prime, and x = y if there are no more to be found. However according to the logic of construction, all one can say is that x ≤ y, because we have no way to decide the question. We can't say anything about whether or not there are any more fermat primes to be discovered, according to the current state of knowledge on the matter.
For the trichotomy principle, one needs a more elaborate example. There are many questions about numbers that haven't been settled one way or another. Here is one of them: A perfect number is a number such as 6 which is the sum of all its factors including 1, but not including the number itself. So 6=1+2+3, and 28=1+2+4+7+14. Are all perfect numbers even? Another one: Is every number a sum of two primes? Nobody knows, nor does anyone have any method which seems likely to solve either of these in the near future.
So two conjectures are that there is no odd perfect number, and that every even number is a sum of two primes.
Let A be one conjecture, and B another one.
Start with the sequence 1/2 + 1/4 + 1/8 + 1/16 + ..., which adds up to 1.
Now go through this sequence and if n is a counter example to conjecture A, keep the nth term in the sequence as it is. If it is a counter example to conjecture B, multiply it by -1. Replace all the other terms by 0.
Define x as the result you get by adding up the numbers in the new sequence. So - x has to be a small number between -1 and 1. By adding more and more terms, you know x to more and more accuracy. Most of the terms are zero, which means that as you go on, you conclude that x must be very close to zero. If you come across a counter example to conjecture A first , you will know that x is larger than zero. If you find a counter example to conjecture B first, you will know that x is smaller than zero. But if neither of those things happen, you can't say that x is zero, or that it is larger than zero, or smaller than zero.
So using constructive logic, no-one can say anything at all about whether x > 0 or x = 0 or x < 0. If this particular case ever were settled, one only needs to construct another example. It doesn't depend on the status of any particular example at a particular moment in time, so long as one can continue to provide examples like this that can't yet be settled or shown to have a solution.
There is a replacement for the trichotomy principle in constructive logic. If you can prove that x ≠ y then you can conclude that x > y or or x < y. This works as a replacement in calculus - you can prove all the necessary theorems, but it is awkward to use.
With my approach of using constructive logic for unending subsequences of large finite numbers, and the logic of discovery ("classical logic" as it is usually called) for the large "seemingly infinite" numbers themselves, one has a stronger replacement.
With my approach, you still don't have the trichotomy principle. But you can say that x ≥ y or x ≤ y. That's because you can imagine continuing any unending sequence beyond the horizon of practical limitations into the seemingly infinite, where it becomes a very large finite sequence that stops somewhere beyond the horizon.
This large finite sequence that goes over the horizon is something discovered. So you can then use the logic of discovery for it, and so it is okay to use the trichotomy principle for it.
This doesn't enable one to assert that x = y or x ≠ y. If there is a difference in the two sequences, there might be no way to decide whether the difference becomes noticeable just before or just beyond the horizon of the practical limitations. But it does let you assert that x ≥ y or x ≤ y. This principle yields short proofs of results in calculus, and is as easy to use as the trichotomy principle. One no longer needs to use the awkward constructive replacement for it.
This new constructive logic extended by adding the rule x ≥ y or x ≤ y, is my own idea. I have developed it as far as a system of axioms for set theory, and theorems as far as all the basic theorems of calculus, but this work isn't yet published anywhere. However the idea of an unending subsequence of a large finite number is a recognised, though somewhat alternative approach. It was developed in detail by a group of researchers in Prague led by Vopenka.
Since there are numbers beyond our horizons, one could imagine perhaps becoming able to construct such numbers some time in the unforeseeable future, at which point one would have further horizons.
To make this a bit more concrete, start with the numbers we can construct in practice, or could construct in the easily foreseeable future. Then we suppose that there are some numbers beyond that. Let's just introduce one, and call it something, say, alpha. We are just giving it a name to give us something to work with. But our reason for thinking this meaningful is because we suppose there really are numbers beyond our horizon.
Then using this new number alpha as given, we can use it's name in mathematical formulae, and so go on and construct larger ones. Since we can go on constructing such formulae endlessly, that makes a new number series with more extensive horizons. But that one will have its limitations too. So let's introduce a new number beta even beyond that one. It will be a basis for a third number series. Obviously we can go on in this way as long as we wish.
We don't know any way of writing down such a number as alpha, by the definition of the type of number it is, as a number beyond any we could write down. But we feel there must be more numbers out there, so as a way to start to think about them, we introduce a name and think of it as applying to one of them.
When calculus is developed in this way, you can use the reciprocals of large finite but seemingly infinite numbers as "seeming infinitesimals". My approach actually has two sizes of seeming infinitesimals (arising from three number series, each more extensive than the next). This gives very concise and simple proofs of many theorems of calculus using the new constructive / classical logic.
The idea of using infinitesimals in mathematics to develop calculus was first introduced by Robinson. His approach introduces them as an extra overlay on top of the usual approach, which is useful because it can shorten proofs of theorems. But the way the infinitesimals are constructed is fairly complex.
With Vopenka's approach, they are introduced as part of the foundations of the theory, and just need an extra axiom or two when setting up the theory.
Robinson's infinitesimals are so constructed that anything you can prove using them, you can prove without them, and vice versa. All theorems (except the ones about infinitesimals themselves) are identical, with or without them. With Vopenka's approach, you have to re-prove all the theorems of calculus again, as there is no guarantee of equivalence of the two approaches. So the infinitesimals in his theory are in a sense simpler conceptually, or from a philosophical point of view - but using them to prove things needs more work initially since without the "star transform" of Robinson's work, you have to prove the basic theorems of calculus all over again.
My approach follows Vopenka's lead, and introduces the seeming infinitesimals into the basic axioms of the theory. So one has to re-prove all the theorems of calculus again.
A principle called induction is often used to prove results about finite numbers.
The theory has two levels of discourse. One way of speaking, about finite numbers, has a method of induction that lets you prove results about all numbers, whether seemingly finite, or seemingly infinite. For instance, you can prove that every number has a unique factorization into primes in the usual way.
Then the unending sequences and seemingly infinite numbers are introduced using new axioms. There is a new principle of induction only over the seemingly finite numbers. For instance, you might prove that every seemingly finite number has a unique factorization into seemingly finite prime numbers.
Finally, bringing the two together, there is a method of transfering some of the results you might have proved only for seemingly finite numbers into the seemingly infinite. If a result doesn't mention the property of seeming finiteness, and holds for all seemingly finite numbers, then you can conclude that it holds for at least some seemingly infinite numbers.
The motivation behind this is that results about finite numbers (as distinct from results about unending sequences of seemingly finite numbers) have to either hold or not for any particular finite sequence. Any statement you can make is decidable in principle, even though it may be well beyond our powers to do so at present. So since the border of seeming finiteness is not sharp, it is impossible for any such property to fail exactly on the border between the seemingly finite and the seemingly infinite.
If the methods are restricted to a particular field of study, the seeming infinitesimals might be actual concrete numbers that one could write down using more concise and powerful methods than are in use in that field. Then one could extend the field by adding alpha and beta as names that can be used in formulae, without introducing the methods needed to get to them from the finite numbers. Then for the purposes of that field, they could count as seemingly infinite. If the field doesn't require enormously large numbers, they might even be comparatively easy to describe, comparable to the enormous numbers mentioned above.
Can one also use the logic of discovery for truly infinite sequences? I think there is an underlying infinity beyond all the practical limitations, and one could in principle make discoveries about it. But I don't know how one could decide what mathematics is appropriate out of the various possibilities. I feel it might even be possible that the description of infinity as having a multitude of sequences of consecutive finite numbers, of varying lengths, may have some validity for the underlying infinity beyond all practical limitations. I feel this because the version of calculus one develops in this way has a certain beauty and conciseness, which mathematicians tend to consider a hallmark of truth. Though I accept that this is a heuristic that doesn't always work and can lead one astray on occasion.
The usefulness of the mathematics doesn't depend on this. However, after working with it for a while, one comes to wonder if it belongs to the realm of things that are discovered.
The mathematics has a static and a dynamic aspect. The finite numbers, large and beyond the horizon, or more ordinary finite ones, have fixed properties. But the constructions by which one generates them are dynamic, in flux, as one constructs larger numbers to place within the endless sequences. Not that they evolve in time particularly, but that one can discover more members of them with time. One wonders if this corresponds to anything in the realm of discovery - perhaps it has a dynamic aspect to it too, and a kind of life to it. Perhaps the platonic mathematics doesn't have to be thought of as completely static, as usually presented.
Something about me and what I do