# Speculative Science

Among the ideas about science, it is this: science examines reasons, proves it is not a hypothesis and speculation refuses to … We will argue that this is a mistake.

In many areas, science is exploring possibilities, offers varied and contradictory attempts to explain, trying ideas, new methods and reasoning and risky. Sometimes she can afford neither to validate nor to test: it engages in what must be called speculation. Is this legitimate? That he participates in the normal functioning of the science?

From our perspective, there is no fundamental difference between a hypothesis and explore and indulge in speculation. For us, it is legitimate for a researcher studying formula and assumptions, provided they are identified as such and no confusion is possible between, on the one hand, the facts or accepted theories, and Moreover, the facts or hypothetical theories. It is in the nature of science to try to understand it and ignore it follows that the assumptions whose consequences are explored carefully – or sometimes just queued to be explored – is a main purposes of scientific research. It is daring to what other researchers have not dared speculator being better than its competitors, we advance. Intellectual recklessness is not to banish from science, it is the normal operating mode!

The idea is not new, and it is defended by example in its own way – that is to say, with an emphasis on falsifiability – Karl Popper1 “bold ideas, unjustified anticipations and speculations are our only way of interpreting nature, our only tool, our only instrument to grasp. We need to use risk for the prize. However, do not really participate in the game of science that those who expose their ideas and take the risk of refutation. ‘

Rather schematically, we must consider two different cases.

If the assumptions – speculation – are made in well established disciplines and based on the concepts and modes specific disciplines in question and practiced for a long time, it gives some interesting discussions and accepted by all. That, although no firm conclusion is drawn. We do not believe that falsifiability of a theory is as important as Popper defends many hypotheses and theories unfalsifiable, now and for long, are taken seriously (in cosmology, about the origin of life, etc.).

If they are made to the boundaries of disciplines or interdisciplinary themes, it is still interesting, but we may succeed not unanimous. Some researchers will feel speculative excesses … and they express their disapproval: This is for example what happens about the anthropic principle, which, according to the sensitivities of astrophysicists, is regarded by them as fully scientific, or opposite, as outside the realm of science. Science has its conservatives who protect abuse, lack of rigor which some scholars indulge, and require more daring to argue.

The boundary between acceptable and abusive speculation is not fixed with perfect precision. It is fixed at the same time as the science progresses, it is the object of discussion among researchers. It is clear, however, that it is better speculation based on facts, theories and scientific reasoning (or attempting to approach scientific reasoning) that speculation based on arbitrary statements uncertain origins, eg from allegedly revealed texts.

Those who accuse science of being closed, defend “official truth” we would not have the right to criticize or to change, are in error. Science feeds the imagination of researchers and assumptions they offer are much more fun, fascinating and varied as those made without withholding on behalf of religious doctrines or various superstitions by those who, by refusing to channel their imagination and the feeding of facts and ideas learned from science, is not limited to that explore assumptions already refuted, poorly constructed, or whose likelihood is extremely low given the lack of rigorous foundations … and often a lack of skill and training.

It is of course more difficult to speculate from a set of theories, concepts and facts that we had to learn, that ramble without railing. The situation today is nothing new, and in all major disciplines, speculation has been widely used. This is true, for example, mathematics, physics, astrophysics and cosmology, biology, economics, psychology, etc..

Let us mathematics we sometimes forget to mention in the list of large speculative disciplines.

Mathematics

For common sense, mathematics is a discipline considered completely rigorous, which was quick to believe that everything is just right – because shown – or false – because proved wrong! The situation is actually much more difficult and complex. All is not simply true or false, and also a word is booked to speak precisely those assertions which we can not know if they are real or only pretend to be. In mathematics, we use the word conjecture, and it is quite reasonable to ask the following equation:

Mathematical speculations = Conjectures

Except in very rare exceptions, no confusion occurs in the mind of a mathematician from a theorem proved, and conjecture. Both state truths, but the theorem is bound by the demonstration to other mathematical truths which serve to support, and no doubt of any kind can not be applied to them (unless the demonstration is wrong). Conjecture in turn is only a “possibility of truth, awaiting validation by demonstration”, or something that contradicted nothing but appearance could in principle be overturned tomorrow.

Arithmetic conjectures

Some conjectures are open for more than two thousand years, here is an example.

Guess the odd perfect numbers

We know even numbers that are equal to the sum of its proper divisors: 6 = 1 +2 +3, 28 = 1 +2 +4 +7 + 14. It is said that 6 and 28 are perfect numbers. Is known in all 47 (February 2010) but it is not known if there are infinitely: we conjecture that yes. More interesting: there are no known odd perfect number, but no one ever has been able to demonstrate that they do not exist. The statement: “There is no odd perfect number” is now considered a conjecture very plausible, but unproven.

It is considered unlikely because, for example it has been established that if n is an odd perfect number, then it has at least 300 digits (no need to look for odd perfect numbers by hand!), Moreover, its

prime factors are at least 75 in number, and at least one of them exceeds 100 million. Some heuristic reasoning – that is to say based on probable judgments (probabilistic nature in general) but not perfectly rigorous – reinforce the idea that the conjecture is true. Still, it is a statement whose status remains uncertain that a statement which we will likely exclude false when a demonstration has been proposed.

This type of situations where mathematicians interested in statements they think true but failed to demonstrate it is rare? No, the mathematical contain thousands of speculative assertions of this kind. Some books make lists and classent2.

Mathematics, contrary to what some people think, is not only a science demonstrations and theories that place because we know all of the objects of which they speak. No, mathematics is also a science exploration, mental constructs unfinished, incomplete formulations sought to clarify doubt and mystery.

All kinds of methods (heuristic, computer, etc.). Allow to formulate new conjectures. Mathematicians speculate and speculation is one of the engines of their action.

Computing that allows massive calculations (and thus a prolonged experimentation scale) leads to some surprising situations, including one that focuses on the so-called champions jumpers. The following conjecture is particularly interesting because nobody can believe is false, at the same time that no one can defend it is true … in the full sense of the mathematician.

Guess champions jumpers

Until the whole 389, the most frequently observed gap between consecutive primes is 2. Then it is 6 and it seems to stay very long. The prime number theorem – which indicates that the density is decreasing – however, implies that 6 does not always champion.

Andrew Odlyzko, Michael Rubinstein and Marek Wolf, leading computational analysis and heuristics, have identified a remarkable law which states that:

The champion was dethroned 6 to 1.7 in 1036 and was then replaced by 30 = 2x3x5.

More generally, the successive champions are products of prime numbers: 2 and 6 = 2 × 3 and 30 = 2 × 3 × 5, then 210 = 2 × 3 × 5 × 7 and 2310 = 2 × 3 × 5 × 7 × 11, and so on.

If we denote by D (n) = 2 × 3 × …. × pn product of primes up to n-th, then D (n) become champion from the number:

N (n) = exp [(2 × 3 × .... × × pn-1 (pn-1) / ln ((pn-1) / (pn-2))]

This gives the table (part) as follows:

n D (n) N (n)

2 6 321

1.70 × 30 March 1036

4210 5.81 × 10428

May 2310 108,656 1.48 ×

6 × 30 030 1.30 10138357

3.56 × 10 6469693230 1074595540317

2.10 × 11 200560490130 102486392448589

Tested using computers conjecture is confirmed.

Well aware that the type of analysis carried out because of a strange science of arithmetic, the authors write, “There is no proof or theorem, only the results of computers, conjecture and interpretation of regularities that observes a physicist. ‘

However, these findings on jumping champions “can not” be accidental, and therefore we are in the strange position of a mathematical statement which no doubt, because it is inconceivable that there has been either accident, but no one can hope to show in the near future, because even the twin primes conjecture – the same type but more basic – resists all attacks.

Infinity

Arithmetic is a field of mathematics where the nature of the objects is not a problem: everyone accepts the idea that there are (at least potentially) integers and their properties are set independently of us and what we want. It does not go well in all areas of mathematics. In set theory, for example, uncertainty and speculation are then of a more profound and more difficult because it is no longer limited to the statements but also focuses on entities and their relationships.

For example (but a large part of set theory is concerned with similar problems) we consider the continuum hypothesis.

At the end of the nineteenth century, the study of the infinite, regarded until then as not belonging to the science becomes, through the temerity of Bolzano and Cantor (who had to endure hostility Leopold Kronecker) a legitimate mathematical topic. This change of status, which does not happen smoothly and without jerks, is a striking example of the expansion of science. Infinity leaves – at least partially – the field of philosophy and religion, where he was to enter what is considered today without any discussion as science.

In other words: speculations on the infinite which is engaged philosophy and religion and does not seem to have learned a little about it (as these speculations were dogmatic and rhetorical games) change nature. They become a topic honorable rigorous progressing rapidly at first, then encounter difficulties, obstacles making some few years or a few decades, other persistent today. The old speculative discussions and producing nothing tangible is now continuing in the context of set theory and the work became recognized and build cumulative knowledge true and deep that nobody discusses the scientific status. Finally, much better than the unfounded arguments or constructions that preceded the passage mathematics can say that every year, the knowledge of the infinite progress.

Among the adopted knowledge, there is the assertion (due to Cantor) that there are infinitely many different kinds of infinities, the lowest levels are called countable infinity (the integers) and continuous infinite (the real numbers, or an infinite straight line). The simple question of whether there is a link between the countable infinite and continuous remains unresolved. The statement: “Any infinite subset of the set of real numbers can be put in bijective correspondence with either all integers or with the set of real numbers (ie: there is no infinity intermediary between integers and real) “is called the continuum hypothesis and denoted HC.

For over a century, we discuss, and if it has been established (the result of Kurt Gödel and Paul Cohen) that the basic axioms of set theory can neither prove that HC is true, or that is false, it does not mean that the issue is resolved, but to look for other axioms, discuss their acceptance, and see if they require HC or its negation. This highly speculative work, appearance and sometimes philosophical about an area of abstraction disconcerting continues and gradually seems to lead (particularly through the work of Hugh Woodin) concluded that HC would be “wrong” . New axioms, a subtle argument defends and leads to accept as natural, were added to the usual axioms of set theory and lead (almost) to the solution attendue3.

Science in this case did not extinguish the speculation, she helped to flourish, she opened a field where imagination, far from being blocked, is encouraged to go further and can do since it produces is not left to the judgment of authorities embodied in institutions or particular individuals, but is subject to a collective and disinterested: the universal mathematician. This universal mathematician, every mathematician particular embodies judge what is worthy of consideration and what does not deserve it, but more often, it is consistent with all other modes of production because judgments are sufficiently clear and explicit for no more arbitrary, no more psychology, no more controversy and no personal conflict does insinuates and role plays. We say that I and simplifies the effects of school authority, nationality play in mathematics, as in all human societies. I admit it, but this game is limited and generally does neither truth nor the interest of major results on which agreement is most often unanimous.

Science here, not only tolerates speculation, but she allows, encourages and provides the means to become bolder, even more crazy.

Talk about the set of all sets, sets that evoke elements could be themselves, consider an infinity of infinities, consider an infinite sequence of infinite and take the limit, etc.. all these ideas (and other more unlikely still) have citizenship and do not cause rejection as the field, its methods and objects were made clear and specific constructive discussions … they are also speculative.

All this is true of the infinite, but could be said about the chance, the notion of truth (as studied mathematical logic). A substantial part of mathematics is made of speculative philosophy or science to say more is just speculation constructive, fruitful and controlled.

What we have just discussed for mathematics is obviously true for other sciences. In quick ideas, all the more daring and risky than the other, the researchers plan and produce scenarios for the origin of life, others to the birth of the universe. They offer models of the universe as a whole, trying to guess what might happen in the very distant future the solar system, the galaxy or even the universe conceived as a whole. They discuss the existence of extraterrestrial life and the likelihood, nature, etc..

Yes, science is open, so it tolerates and encourages speculation that nourish and even when they lead to non-testable hypotheses, are nevertheless not work with the scientific value and are accepted by almost all . The slight hesitations and disagreements at the boundaries of disciplines on topics such as the anthropic principle, the origin of life, extraterrestrial life, parallel universes, theories of everything, and a few others that some may be reluctant to consider legitimate show that precisely the rationale that builds science is not a foregone conclusion.

Math for fun

An inventory of curiosities

Jean-Paul Delahaye

Publisher: For Science Collection, February 2010, 208 pages, 25 €

Mathematics is easy and engaging there is a pleasure. The easiest proof comes from the music that is always, in one way or another, a set of abstract mathematical nature, which makes everyone feel the infinite beauty of pure forms and immaterial forms are precisely the concern mathematician. Geometric and typographic arts, card games, games with dominoes or checkers with the social and political life and its subtle strategies, trade, all these activities are mathematics and often provide satisfactions … even those who claim not to like math and be “zero”.

The aim of this book is to persuade readers who are not already, that mathematics can not be reduced – thankfully – to what we learn in school, and everywhere present, they are a source of joy and fulfillment for those who know how to spend a little attention and playfulness.

The five main themes of the book are: Arts and mathematics Geometries fun, Games, Numbers, Puzzles and riddles.

Compounds from the articles under “Logic and computation” that appear each month in the journal Science for the 22 chapters of this book can be read in any order you like, and even partially not focusing that ‘the figures and boxes … if that is your pleasure.