How We Accept New Ideas of non diophantine
However, in spite of all evident facts that prove existence and usage of non-Diophantine arithmetics, people’s conservatism, inertia, and bias prevent them from empowering themselves with this tool of cognition and practical activity. An important regularity of society is that its life is based on a multiple stereotypes. This is also true for science (Burgin and Onoprienko, 1996). There are many historical examples of such situations.
One of the most notorious is from the history of the discovery of non-Euclidean geometries, which was mentioned before. Not only laymen did not want even to listen about this outstanding achievement of human mind, but even well-known and authoritative mathematicians opposed this new theory. For example, Ostrogradsky, who was a contemporary of Lobachevsky and was considered the best Russian mathematician at that
time, wrote a withering review on the work of Lobachevsky on non-Euclidean geometries. In addition to this, a pamphlet on Lobachevsky was published in the most important Russian newspaper of that time. The main thesis of this pamphlet was that some people come from the province, call themselves mathematicians, and write very clumsy and awkward texts about such things that are impossible because they are never possible.
Also Read : More Properties of non diaphantine arithmetic
We know that Kant claimed that (Euclidean) geometry is given to people apriory, i.e., without special learning. However, one might say that Kant was a philosopher and not an expert in mathematics. Besides, when Kant wrote about geometry, non-Euclidean geometries were not yet built. Only some people suggested hypotheses that other geometries might exist. However, William Rowan Hamilton (1805-1865), certainly one of the outstanding
mathematicians of the 19th century, expressed similar consideration in 1837 when the works
of Lobachevsky (1829) and Boyai (1932) had been published. Hamilton said:
“No candid and intelligent person can doubt the truth of the chief properties of Parallel Lines, as set forth by Euclid in his Elements, two thousand years ago; though he may well desire to see them treated in a clearer and better method. The doctrine involves no obscurity nor confusion of thought, and leaves in the mind no reasonable ground for doubt, although ingenuity may usefully be exercised in improving the plan of the argument.”
Even in 1883, another outstanding mathematician Arthur Cayley (1821-1895) in his presidential address to the British Association for the Advancement of Science affirmed:
“My own view is that Euclid’s twelfth axiom [usually called the fifth or parallel axiom] in Pfayfair’s form of it does not need demonstration, but is part of our notion of space, of the physical space of our experience…”
Moreover, it is known (Kline, 1980) that even in 1869 and 1870 the noted French mathematician Joseph Bertran (1822-1900) published in Comptes Rendus (Bulletin) of the Paris Academy of Sciences his works in which he disproved non-Euclidean geometries.
Outstanding achievements of the great French mathematician Évariste Galois (1811-
1832) give another example of such situation. As writes E.T. Bell (1965), “in all the history of science there is no completer example of the triumph of crass stupidity over untamable genius than is afforded by the too brief life of Évariste Galois.”
The first time Galois submitted his fundamental discoveries to the French Academy of Sciences, Cauchy, who was the leading French mathematician of that time, promised to present this, but he forgot and even lost the abstract of the works. The second time Galois submitted his epochal papers on the theory of algebraic equations. These ideas later gave birth to the modern algebra, in which structures were studied instead of equations. However, these works were also lost because nobody understand either the essence of Galois discoveries or even their importance. Only by a mere chance his works were published long after his death.
Still another example of the stereotypes that hinder acceptance of highly original and important theories and ideas is the attitude to the works of the outstanding German logician Gotlob Frege (1848-1925). Grattan-Guinness writes in his History of the Mathematical Sciences (1998):
“Neglect [of Frege] during his lifetime is normally explained as due to his unusual notation … but I learnt it in 20 minutes and I cannot see why his contemporaries would have needed longer.”
The life and works of such great English physicist as Oliver Heaviside (1850-1925) give one more example and we can find a lot more.
What concerns non-Diophantine arithmetics, here are we describe only two situations. The first one is related to one of the well-known contemporary physicists. When somebody came to him with dubious results in theoretical physics, the physicist said that that results are not possible and continued:
“You might as well claim that you can prove that 2 + 2 = 3. If you think you can do that, don’t waste my time, go to the market and buy two $2 items and try to pay for them with $3.”
Some time after this episode, this physicist was informed about non-Diophantine arithmetics, in which it is possible that 2 + 2 = 3, and given an example when it was possible “ to go to the market and buy two $2 items and try to pay for them with $3.” This gives an experimental proof that it is possible that 2 + 2 = 3.
The reaction of the physicist was very characteristic. Although he consented that it is possible to “go to the market, buy two $2 items, and pay for them with $3,” he rejected the possibility of other arithmetics. Here is exactly what he responded:
“I just checked on my fingers: 2 + 2 still = 4. Therefore, I have proven experimentally that the theory [of non-Diophantine arithmetics] is wrong.”
Then he added, “Mathematicians are fond of calling an apple a banana.”
At first, let us look at his “as-if proof.” In a same way, some people “prove” that there are neither atoms nor subatomic particles. They say, “I just looked around and did not see any atoms. Therefore, I have proven experimentally that the theory [of atoms] is wrong.”
Then such a hypothetical person may add, “Physicists are fond of inventing things that do not exist.”
A similar situation with “as-if proofs” is described in the following joke, which is aversion of the one from (Polya, 1956). The joke “How a mathematician, physicist, engineer, and computer scientist prove that all odd integers are prime.”
The mathematician says, ” Well, 1 is prime, 3 is prime, 5 is prime, and by induction, we have that all the odd integers are prime.”
The physicist then says, “I’m not sure of the validity of your proof, but I think I’ll try to prove it by experiment.” He continues, “Well, 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is … uh, 9 is an experimental error, 11 is prime, 13 is prime… Well, it seems that you’re right.”
The engineer then says, “Well, actually, I’m not sure of your answer either. Let’s see… 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is …, 9 is …, well if you approximate, 9 is prime,
11 is prime, 13 is prime… Well, it does seem right.”
Not to be outdone, the computer scientist comes along and says “Well, you two have got the right idea, but you have end up taking too long doing it. I’ve just whipped up a program to REALLY go and prove it…” He goes over to his terminal and runs his program. Reading the output on the screen he says, “3 is prime, 5 is prime, 7 is prime, 7 is prime, 7 is prime, 7 is prime, 7 is prime….”
In another version of the joke, the physicist then says, “I will prove that all odd integers are prime by experiment. Well, 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is … uh, 9 is equal 3 times 3 and as 3 is prime, we may, as well, call 9 prime, 11 is prime, 13 is prime… Well, the experiment proves the claim.”
What concerns the second statement of the physicist about mathematicians, he also makes a mistake based on an inherent reductionist intentions of the majority of physicists – they try to use as little terms as possible. However, imagine if mathematicians (and physicists) instead of speaking about groups, in all cases use the term “a set with a binary operation that is associative, called multiplication, and for which an element e exists, which is called the unit and which is being multiplied by any other element a it yields a, and for any element a in this set there is another element, which is called by these “pernicious” mathematicians the inverse of a, such that being multiplied by a it yields e.”
Another situation with the theory of non-Diophantine arithmetics was even more striking. One man said to the author:
“I don’t know your theory. I will not read anything about it. However, I completely disagree with you.”
Thus, we see that non-Diophantine arithmetics provide a mathematical base for a variety of ideas and situations in different areas, but it very difficult for people to understand even a possibility of existence for non-Diophantine arithmetics, not speaking about their essence and utilization.
It is also necessary to note that in (Burgin, 1977; 1987) non-Diophantine arithmetics are defined in a constructive way by a generative schema. Thus, it is interesting to develop logical description of these arithmetics and to give their axiomatic characterization.