In details

Symmetry in Mathematics IV


It is not paradoxical to say that in our moments
more theoretical
we may be closer to
our most practical applications.

A. N. Whitehead

… We can argue that, since only two symbols, the 0 it's the 1, people, for now, our universe, it is fundamental to decide if 1 + 1 It would be a new entity. Moreover, the same question applies to cases 0 + 1, 0 '1, 1' 1, 0 '0 and 0 + 0: will new ones also be generated here? …

… We have already discussed in previous columns the cases where 1 +… + 1 Can be 0. They are the finite structures of numbers. We now begin the discussion of the case where 1 +… + 1 never is 0, whatever the number of plots in this sum, that is, the case of the real body…

Imagine a number structure where generator 1 never generates 0 in addition is a nontrivial attitude. This imagination immediately raises the problem of inventing the right symbols to represent an “infinite” number of numbers. That is, we need a symbol for 1 + 1, for 1 + 1 + 1, etc. We are immediately faced with the most important idea of ​​mathematics: the idea of infinite. How to get used to this idea? There is nothing in our daily lives that inspires us and that accommodates our spirit to this idea. On the contrary, it is this idea that seems to shed light on the many puzzles of our practical life. But the first problem we have to face, if we like order in ideas, is the problem of representing an infinite number of numbers. This problem is far from easy, so much so that humanity only found a satisfactory solution for it around the year 1000 AD.

It was in India and Arabia that this problem seems to have been resolved. The solution is the Positional System where the symbol 0 has a role of fundamental importance. This invention is comparable in terms of unleashing powerful forces of progress to the computer revolution of the 1980s. Basic calculus is no longer an obstacle to the mathematical imagination. The Positional System could provide an efficient method of generating and representing an unlimited number of numbers from a handful of ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The following number no more representation problems: it is 10, where 0 occupies a position forcing 1 to occupy a new position, thus always allowing the same symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, indicate a number “ten times greater” than generator 1, or any other number of the form 1 + 1 + 1 +… +1, whatever the number of installments equal to 1.

However, this advance was only in the field of numerical computing, but not least. The most spectacular advance was to come in the field of conceptualization. The Positional System suggested that an infinite digit representation (0.111…) could correspond to a number in the same way as a finite representation such as 2.3045. At this point we are faced with three ideas of colossal importance: in addition to the idea that there are infinite numbers, which we mentioned earlier, we come across the idea that there are numbers that require an infinite decimal representation and the idea that there are infinitely small numbers. . Here is a pillar of the idea of ​​symmetry that permeates contemporary science. Symmetrically to the idea of ​​infinite numbers, hence the idea of ​​infinitely large numbers, we come across the idea of ​​infinitely small numbers that is inseparable from the idea of ​​numbers with infinite decimal representation.

Not surprisingly, such a structure has infinite wealth as well. It contains, for example, a representation for an infinite loop. It is easy to find finite cycles in nature such as day and night, seasons, biological cycles, etc., but infinite cycles do not seem to exist in the real world. Paradoxically, the idea of ​​infinite cycle, associated with the numbers 1, 1 + 1, 1 + 1 + 1,… seems natural to us and causes us an irresistible attraction. What is surprising is that such a structure can be useful in solving our practical and theoretical problems, even in the face of our inability to detect infinite cycles in nature. The finite world finds fantastic applications of the idea of ​​infinity.

Following a certain order of ideas, we need to solve a problem: Why does the infinite loop 1, 1 + 1, 1 + 1 + 1,… have a beginning and no end? In our imagination this does not seem to be a mortal defect, but it is undoubtedly an intriguing feature: it is therefore asymmetrical. If we create “your left side” we eliminate this asymmetry. Thus, we incorporate the negative symmetric numbers generated by 1 and have:

… , -(1+1+1), -(1+1), -1, 1, 1+1, 1+1+1,…

But between negatives and positives appeared a "vacuum" that could be filled by a number that was "neutral," that is, neither positive nor negative. This small asymmetry can be eliminated by 0. We are then left with a more symmetrical infinite cycle that has no beginning and no end, and is infinite:

… , -(1+1+1), -(1+1), -1, 0, 1, 1+1, 1+1+1,…

Now our mind seems to have reached a state of balance. The numerical representation above seems to have encompassed as much of our imagination as possible to represent infinite cycles. Until a new problem challenges us we may be satisfied with this representation of the possible numbers. At least for the numbers generated by 1. This seems to be the maximum structure generated by 1 when our imagination demands that never 1 + 1 + 1 +… +1 be 0.

We had mentioned above the idea of ​​infinitely small numbers. Where does this imagination come from? Again, it comes from the solution of one more intriguing remark: Why would there be only infinitely large numbers like the ones we get in the sequence above? That is, 1 generates infinite numbers by addition and thus generates infinitely large numbers. Could we not imagine infinitely small numbers as well? Whose latter would be generated by whom?

After a few moments of rigorous introspection, we can see that we are really facing a new challenge, a new asymmetry. We have found that it is not satisfactory to accept a numerical structure that includes numbers that can only grow. We have detected a lack of numbers that may also decrease. This asymmetry needs to be corrected, otherwise we will have to live with a restlessness that will undoubtedly become dissatisfaction. But how to solve this problem further?

Once again the Positional System will show its immense power. Very simple arguments can be triggered to unravel this hidden symmetry of the infinitely small by making a counterpoint to the infinitely large. After all, isn't this the intuition that suggests the experience of living with Physical Nature?

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