"Sorry, my keyboard has only the English letter l, not the Polish l_bar."
Copy-pasting doesn't work either? Other people somehow managed to spell it right.
Not that I cared, you can call me Fido if you like. What's more important to me is your comment's content.
"Unfortunately already your Fig. 1 seem to indicate that you didn't reveal, tackle, and solve difficult problems that cannot be seen by professionals who are blind because of their indoctrination."
Now that's what I call judging books by their covers (or figures in this case). If you stopped at the first figure, then you missed a lot, and I'm not sure you're entitled to judge my essay at all in that case, because your opinion will be ignorant anyway.
Please tell me, however: how did you come to your conclusion (once more) that I "didn't reveal, tackle, and solve difficult problems" just from (not) reading these measly 9 pages of mine? 9 pages is quite limiting (even Einstein had 24 pages for his "On the Electrodynamics of Moving Bodies" ;-J ). I couldn't fit in there much of my ideas on the subject, since each of the great FQXi questions induced a flow of creativity that couldn't fit on 20 pages each! (That's why I'm planning to make it somewhere else soon.) So I don't think it is in any way a representative sample of what difficult problems I'm capable of solving. (Especially that solving difficult problems was not the subject of this contest; it's about the "mysterious" connections between Math and Physics.)
And if solving the problem which has not been solved by Newton, Leibniz, nor anyone after them throughout the next 300 years (a little sample of which I revealed in my essay) is not "difficult" enough to you, then please tell me: what problems do you consider "difficult"? I'd be very happy to try my hands on those too (if only I find some free time for that exercise, since I'm quite busy already with my own research). I'm also curious what difficult problems have you solved.
"Elsewhere I objected to your opinion that numbers are pebbles."
And I already answered that question, unless you didn't notice. So why are you still nitpicking that particular word? Couldn't you find anything more serious to be nitpicked in my essay? (Oh wait... right, you stopped after the first figure...) In case you missed my previous reply, or it wasn't clear enough to you, here's one more explanation of this particular problem:
I used the word "pebble" in the context of numbers, since this was the first notion from which we derived the concept of number anyway. It all started from counting pebbles (Latin "calculi"), or using pebbles to represent the amounts of other things (e.g. sheep). It is related to the first abstraction of numbers: that about counting. Actually, counting is more basic than the concept of a number, and it doesn't require numbers. It is based on comparing the groups of individual objects, one by one, to find out if they are the same. This allows us to transfer the knowledge of the amount from one group to the other (equivalence). But it doesn't tell us yet how many objects are there, until we construct a reference set of such groups, ordered increasingly, so that we could use counting to compare other groups of objects with the groups in this reference set. This is how natural numbers come to life.
But, as I already said, I'm perfectly aware that natural numbers is only a good start, and it won't suffice for long. Our theory about numbers has evolved many times since then, and it evolves still (which I see as a great example that we actually not invent numbers, but discover them). We noticed that there could be fractions of a whole, which have to be counted somehow too, so we introduced rational numbers into our number system. Nevertheless (as I show on my website), rational numbers don't introduce anything new into the number system yet: they're just natural numbers in disguise, since you can always find a common denominator (or common unit of measure) for all the fractions you use and convert all these fractions to natural numbers, just based on a different unit. It is a known notion in number theory that if some theorem has been proven for natural numbers, it can be usually proven for rational numbers as well, and vice versa. This is also responsible for the fact that rational numbers have the same order of countability than natural numbers (in Cantor's sense), the famous "aleph 0", or "countable infinity" (I prefer calling them "discrete", which is more descriptive).
The first type of numbers which actually introduced something new, were irrational numbers, because together with rational numbers they make continuum. And here's where you can object to my statement that every number is always expressed in relation to some unit. But I hold to my statement, and here's why: Without having your unit defined, you cannot actually measure your number in question. That's because the information stored in the number and expressed by it, is always relative to the unit. It tells how many times you have to repeat that unit to get the length of your number, and how much do you need to rotate it and in which directions to produce it. This holds for irrational numbers, complex numbers, quaternions, octonions, sedenions etc. I haven't found any number yet which wouldn't hold to this law, and I cannot imagine how could this law be broken at all, since the relativity of numbers to their unit is inherent to their very definition I use. The unit is what all these numbers are constructed in one way or another.
This also hold for non-algebraic numbers, such as pi, since you can always construct a unit circle, and the number representing the length of its circumference will be in a certain and constant relation to its unit radius (namely 2 pi). The actual "algorithm" you use to actually measure this length is irrelevant to this statement, because there are better and worse algorithms. E.g. one algorithm would measure the unit at once to give its numerical name as "1", and the other would fall into infinite loop of generating nines (1/9ths of the previous unit in a geometric progression). That's what Zeno already noticed 2000 years ago, and it's unfortunate that people akin to you see Zeno's musings as a fallacy of Ancient Greek mathematicians who "hadn't known our modern notion of limit". (True, they didn't call it by that name, but they knew it all well - see Archimedes and his method of exhaustion for example.) Zeno's paradoxes are paradoxes only to those who confuse numbers with their numerical representations. I see them myself as a type of Zen's koans which were designed to make the followers of digital notation to see its inherent limitations. There are numbers which cannot be expressed digitally, since the algorithms used to generate digits, based on geometric series, don't halt for these numbers, producing digits ad infinitum without any repeating patterns.
But one can escape the paradox easily: by noticing that Geometry, and its "programming language" which is Algebra, are still capable of expressing these numbers and measuring them in relation to the unit. The paradox was the Mathematics' way to tell us that our assumption was wrong, and not every number can be expressed through those numerical algorithms in a finite way. But there are other algorithms beyond that which we can use. We just need to figure out that seeing multiplication as repeated addition, and exponentiation as repeated multiplication, was only the first approximation of what they really are. This is just a special case of a more general concept, which is geometric similarity and proportion. Multiplication, exponentiation and logarithms are all based on this notion. One just need to spot it. Then one can use geometric similarity to find ratios of irrational numbers (so to speak) with respect to the unit. The relation to the unit is more subtle than just simple repetition of whole parts or fractions, but it is still there, and will always be.
But we need to learn from our past mistakes and generalize our understanding of numbers once more: to see that they express not only magnitudes, but also directions, and that they are something even more: operators which can perform transformations from one state to another. It means that they have some dynamical aspect in them, and this can be used to merge number theory with differential calculus.
"Initially, you correctly followed Euclid who explained the role of the unit 'one'."
If you read the rest of my paper, you would know that I'm still with Euclid on that. I just go further with it, by observing that numbers have not only magnitudes (as Euclid's line segments), but also directions (the segment has its starting and its ending point, and it is being "traveled" one particular way) which can be expressed in relation to some unit. Traditional Euclidean geometry works as if directions were unimportant and magnitudes were taken as absolute values, so it is a subset of what I use for expressing numbers.
On the other hand, I strongly object to the idea of expressing numbers as mere sets. Numbers are not sets - they are contained in sets! Sets don't have magnitudes (not to mention directions), so their use for representing numbers can takes us only so far. I wonder how can one express a transcendental number such as pi with sets, derived from the first principles (axioms of the set theory). Cauchy sequences does not appeal to me, since approaching a number, or proving that it exists somewhere on the number line, is not the same as expressing the number itself (not a constructive proof, if you don't have the exact object in question in your hand).
"However in geometry, the unit was to be understand as a distance, in other words a measure, not a 3D pebble on the abacus"
I never said what you say anywhere. You're putting into my mouth what didn't come from them, a typical strawman argument (again!) which is considered a logical fallacy. What I said is that every number is expressed in relation to some unit and therefore arithmetics can be "reduced" to operations applied to such units. The unit can be anything, and it is indivisible "by definition", so it can be seen as a pebble. It can be seen as the Euclidean line segment, as you said, and I don't object to that, but it can also be a unit square tile, a unit cube, an actual pebble, a litre, a second, a volt, a kilogram, or some more abstract ideas, since numbers don't actually care what that unit is - they only care about the transformation of that unit into something else, by repeating this unit a certain number of times in a particular direction. (And in before your nitpicking, what I mean by "repeating a certain number of times" is the more general sense of geometric proportion, which can be irrational, not just repeating whole "pebbles", as you trying to impute me to have something to object later).
