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Role of mathematics in science

It has been four centuries since Galileo Galilei discovered the scientific method as we know it. What were the main breakthroughs that allowed him to switch mankind from the pre-scientific mode of thinking to the scientific mode of thinking?

Time-dependence controlled by maths

No doubt, one of his most important contributions was the discovery that mathematics and quantitative thinking can be applied and should be applied to dynamical processes. As Steven Weinberg said, it was Galileo who put time into physics. No one before him had this idea: the closest thing people knew was static geometry, a gift of ancient Greece, that can be viewed as the oldest branch of physics describing perfectly solid and static objects in a certain limit, although nowadays we usually re-interpret the history and include the Greek discovery to be a part of mathematics that has nothing to do with natural sciences. However, the Greeks didn't respect our modern boundaries between physics and mathematics and they would surely have no difficulties to be counted as physicists.

In one of Galileo's key experiments, he had to determine whether the acceleration by gravity increases the velocity by the same amount per unit time or unit distance. His experiments with inclined planes in Pisa are legendary. The answer to this question was, of course, that the velocity increases under the influence of gravity by the same amount per unit time.

Refining the theories

Galileo drew the road map to study Nature ever more rationally, relying on ever more careful quantitative arguments and measurements and using ever more sophisticated mathematical structures. Isaac Newton had to discover the calculus in order to describe the motion of apples and celestial bodies. Others had to invent partial differential equations, Lie groups, Hilbert spaces, path integrals, RG flows, conformal field theory, and many other things that were necessary for a proper quantitative understanding of various physical systems.




As our understanding of various physical systems got better, we were able to make increasingly non-trivial and sophisticated predictions with ever smaller amounts of input information. We could understand and predict much more than what we could directly see with our eyes. As physics was making further progress, the gap between these two categories kept on increasing. Today, we understand the internal architecture of cells, atoms, as well as protons; we know how the Sun produces the heat and how the Universe has been expanding for 13.7 billion years even though all these insights, and millions of others, are clearly well beyond what we can experience directly.

Although there are many fields of science where abstract reasoning is not too important, the role of pure thought in theoretical physics has surely increased, too. Theories were inevitably getting more abstract. The power of mathematics - a tool that allows us to create very indirect yet robust arguments - is in fact known in many other fields such as genetics. It has become normal to reconstruct evolution of DNA by a careful statistical analysis of the DNA code or to study cancer with math. Some critical insights in physics such as special theory of relativity and general theory of relativity were essentially found by pure thought, requiring no new experiments whatsoever. These theories were ultimately proven by experiments but it is fair to say that e.g. the theoretical foundation of general relativity remains more important than the particular experiments that have verified some of its predictions.

Thinking remains a heresy

Nevertheless, 384 years after Galileo famously wrote that

  • Mathematics is the language in which God wrote the Universe,
some people - including some men who have even been awarded PhD degrees in physics - completely misunderstand what science is all about. The owner of the blog called "Not Even Wrong" is a textbook example. He attacks Prof Mark Srednicki, the chairman of physics at UCSB, for saying obvious things that every physicist understands - namely that the task for physics is to find out what is true i.e. correct and what is not: whether the result is easy to see or hard to see - and whether a theory is compatible with some philosophical dogmas - can't influence our judgments about the validity of a theory.

In Lenny Susskind's words, it would be very foolish to throw away the right answer because of its tension with an arbitrary definition of "science". What is needed for something to be science is to be falsifiable in principle: when it is, it is science, and whether it is easy or hard to actually falsify the theory in practice can't play any major role in the search for the right answers.

"What I can't see can't exist"

Mr Woit thinks that if someone uses the word "true" without having a doable experiment, he must surely be a member of a religious church. Some of his readers delight us with their post-modern "wisdom" that the truth doesn't exist and cannot exist.

Mark Srednicki is a particle physicist who has made very important contributions to the question of open string tachyon condensation in string theory (the first "bulk" example) but who may nevertheless be considered to be a general particle physicist by many people because this is where he has made most of his other important contributions. Well, it shouldn't be surprising because this dispute has nothing specifically to do with string theory either. It is a dispute between people like Mark Srednicki who know what high-energy physics actually is and who have been doing it for years on one side, and science-haters like Peter Woit who only know what science is from their insane twisted anti-theoretical ideological rants on the other side.

Srednicki of course talks about quantum gravity - something that has always been a task primarily for the theorists. Everyone whose IQ exceeds 90 and who is given a five-minute explanation what phenomena quantum gravity studies should understand that it is probably very difficult to find experiments that would test the relevant phenomena directly. It follows that theorists' work in quantum gravity is destined to be more important than experimenters' work. It has always been the case, it is still the case, it will probably be the case in the future, and only very limited people can misunderstand why.
The most current discussion between scientists and laymen can be found on this Clifford Johnson's page. See also a routine addition of mine.
Chicken Little Society has been around for centuries

Peter Woit and his comparably "reasonable" readers would apparently like to ban any science whose impact they can't see with their eyes or understand with their brains. What incredible [beep] these people are! Let me enumerate a few examples of criticisms of science in the past that was virtually identical to their present criticism ("it can't be seen so it must be bad"), in order to demonstrate that a certain kind of human stupidity simply can't be eradicated:
  • Columbus was criticized for his plans to try the Western route to India because this reasoning based on the round shape of Earth was pure theory but it would surely be a waste of sailors in practice
  • Charles Darwin has been criticized by creationists because his theory was "untestable": we can't re-run evolution in the lab from the scratch, they still say; Darwinism and string theory are analogous
  • The promoters of the atomic theory in statistical physics and chemistry were criticized for their "pure theory" because it was "obvious" that no one could have ever seen something as small as the atom
  • Chemists were criticized by philosopher Auguste Comte for their attempts to talk about the chemical composition of celestial bodies: this question was forever inaccessible to humans because we couldn't travel into the Sun; it took 7 years only before Comte was proven wrong
  • Alfred Wegener was humiliated for his theory of continental drift because no one can apparently ever move the continents and his reasoning was just a mad theory based on patterns (shape of Africa and America and similarities of plants and fossils, for example) that couldn't correspond to the material world
  • 2,000 Nazi physicists told Einstein that he was wrong as his theories of relativity had nothing to do with the work they could do in their labs (besides his "wrong" race)
  • Mainstream quantum mechanicians were criticized for their statement that only probabilistic predictions were possible; some critics thought it would always be impossible to decide whether there could exist hidden variables
  • The quark theory was criticized because it deals with abstract objects - quarks and gluons - that can never be seen in isolation, and are thus untestable
  • Pierre-Marie Robitaille criticizes the Big Bang theory because it deals with remote (cosmic) sources that can't be created in the lab; the origin of the CMB must surely be in the oceans because the Earth is more testable
I hope that the reader believes me that one could find dozens of other, and probably much better, examples in which science was attacked by the omnipresent Chicken Little Society.

Criticism without content has no value

In all these cases, the critics were wrong. But even if they were not wrong in a similar situation, this kind of criticism was just pure garbage in all the cases. A sensible person simply can't criticize a scientific direction just because it hasn't yet been proven or because it is difficult to prove it. The whole goal of science is to work on theories that have not yet been proven. The only way to criticize a theory is to show that it is wrong and why it is wrong. Whoever criticizes a theory just because it doesn't agree with his emotions and philosophical preconceptions is a bigot.

Whoever thinks that the key adjectives in science, namely "right" and "wrong", should be replaced by some completely different labels that should become primary blatantly contradicts the very general purpose of science: the search for the right answers.

"Mathematical arguments and logic are illegitimate again"

People like Peter Woit simply deny all of science, much like some of their predecessors in the list above. They want to return us before Galileo to the era in which it was not allowed to use any sophisticated and careful mathematical arguments. They apparently don't believe that mathematics plays any role in physics. Scientists could only be allowed to work on their science if they could show the results so that literally everyone - including people like Woit himself - would immediately see them. But this is not the environment in which science could work because science must first work on the hypotheses that are not quite obvious, and only after the work is completed, the results may become manifest.

This dispute is certainly not about string theory only. It is about the whole scientific method for which a careful, cool, quantitative, and mathematics-based reasoning has been more important - at least in the last four centuries - than some philosophical preconceptions restricting what kind of science is allowed and what kind of science is not. In science, a calculation is simply much more relevant than a propagandistic label attached to a theory. At the fundamental level, physics is known to satisfy mathematically solid laws even though a similar assumption could be misleading in other sciences.

Truth vs profit

Nature doesn't care whether Her secrets can be easily discovered or not. If we look at the history of science, it is very easy to see that some secrets were easy to be found while others were not. Predictions of some theories could have been easily and accurately measured, predictions of other theories could not.

It is a rationally unjustifiable bias to prefer theories that are easy to test. In the case of quantum gravity, because of the very definition of the task, we always expected that it would probably be difficult to experimentally test anything about it in any foreseeable future. This is not a new insight. It has been an inherent feature of quantum gravity since the very beginning. For example, the typical distance scales are likely to be as small as 10^{-35} meters - the Planck scale. No sane person could have ever thought that it would be trivial to design experiments that measure what happens at this length scale.

Nevertheless, we can show that at this distance (or a longer one), inherently quantum gravitational effects have to become important. We have made a huge progress in our understanding what happens in this regime.

For example, we know that black holes have thermodynamical properties. When they evaporate, the information is preserved while causality is violated by exponentially small effects. Topology of space can change and different geometries may be physically indistinguishable. Theories of gravity in AdS spaces are equivalent to non-gravitational theories defined at the boundary. These are not philosophical clichés but rather popular summaries of very accurate, rigorous, and robust analyses of various systems that almost certainly belong to the same "universality class" as objects in our Universe.

Direct experiments are not the only tools of science

If we have no easy experiments, it doesn't mean that we should give up. The history of science is literally flooded with examples of correct theories that have been found with a minimal help of new experiments - even though the members of a certain religious cult find this fact highly inconvenient. We don't need to change anything about the scientific method and we shouldn't change anything. It is clear that what the scientific community has to do is to take the best tools we have and deal with them in the most rational and careful way to learn as much as we can. It is very clear that in the case of quantum gravity, careful mathematical arguments and calculations are the key to success and we have already seen quite a lot of this success.

There are dozens of other reasons why the attack by people like Peter Woit is completely irrational. They will tell you that if we find the right answers with the help of careful mathematical reasoning and without any new experiments, it must surely be a religion. But they never realize - and they never admit - that their statement that string theory is not the right direction how to complete the theory of fundamental particles and forces is at least the same religion because they have no experimental evidence for their statement either.

Well, I think that every sane scientist would choose a direction that is justified by robust mathematical arguments and that reproduces all previous theories rather than a direction that is only justified by the hatred of dozens of silly and biased people.

And that's the memo.

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