Physics and Mathematics

The traditional view is that physics and mathematics are quite different. However, many scientists, especially working in both fields, find that there is no big difference between the two fields. It is a matter of degree, of emphasis, not an absolute difference. After all, mathematics and physics coevolved. For mathematics to progress you actually need new ideas and plenty of room for creativity. Mathematicians should not isolate themselves. They should not cut themselves off from rich sources of new ideas.

Physics describes the universe and depends on experiment and observation. The particular laws that govern our universe – whether Newton’s laws of motion or Standard Model of particle physics – must be determined empirically and then asserted like axioms that cannot be logically proved, merely verified.

Mathematics, in contrast, is somehow independent of the universe. Results and theorems, such as the properties of the integers and real numbers, do not depend in any way on the particular nature of reality in which we find ourselves. Mathematical truths would be true in any universes.

Yet both fields are similar. In physics and indeed in science generally, scientists compress their experimental observations into scientific laws. They often show how their observations can be deduced from these laws. In mathematics, too, something like this happens – mathematicians compress their computational experiments into mathematical axiom, and they then show how to deduce theorems from these axioms.

An emerging field of science is experimental mathematics. In this area there are many similarities: the discovery of new mathematical results by looking at many examples using a computer. Whereas this approach is not persuasive as a short proof, it can be more convincing that a long and extremely complicated proof, and for some purposes it is quite sufficient. Extensive computer calculations can be extremely persuasive, but do they render proof unnecessary? Yes and no. In fact, they provide a different kind of evidence. In important situations both kinds of evidence are required, as proofs may be flawed, and computer searchers may have the bad luck to stop just before encountering a counterexample that disproves the conjectured result.

Mathematics differs from physics that is truly empirical but perhaps is not as different as most people tend to think. A Hungarian-born scientist Imre Lakatos came up with an expression quasi-empirical, which means that even though there are no true experiments that can be carried out in mathematics, something similar does take place. Some conjectures are arrived at experimentally, by noting empirically what is true for certain sets of numbers. Some conjectures have not been proved yet, but verified to a certain degree.

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Big Bang

The Big Bang theory isn’t about the bang itself but about what happened after the bang. By doing a lot of math and watching carefully what goes on in particle accelerators, scientists believe they can look back to 10^-43 seconds after the moment of creation, when the universe was still so small that you would have needed a microscope to find it. Most of what we know about the early moments of the universe is thanks to an idea called inflation theory first propounded in 1979 by a junior particle physicist, then at Stanford named Alan Guth. He would probably never have had his great theory except that he happened to attend a lecture on the Big Bang given by Robert Dicke. The lecture inspired Guth to take an interest in cosmology, and in particular in the birth of the universe.

The eventual result was the inflation theory, which holds that a fraction of a moment after the dawn of creation, the universe underwent a sudden dramatic expansion. It inflated − in effect ran away with itself, doubling in size every 10 ^-34 seconds. Inflation theory explains the ripples and eddies that make our universe possible. Without it, there would be no clumps of matter and thus no stars, just drifting gas and everlasting darkness.

According to Guth’s theory, at one ten-millionth of a trillionth of a trillionth of a trillionth of a second, gravity emerged. After another ludicrously brief interval it was joined by electromagnetism and the strong and weak nuclear forces. These were joined an instant later by swarms of elementary particles. From nothing at all, suddenly there were swarms of photons, protons, electrons, neutrons, and much else − between 10⁷⁹ and 10⁸⁹ of each. In a single instant there was a vast universe − at least a hundred billion light-years across and perfectly arrayed for the creation of stars, galaxies, and other complex systems. What is extraordinary is how well it turned out for us.

This is one reason that some experts believe there may have been many other big bangs, perhaps trillions and trillions of them, spread through the mighty span of eternity, and that the reason we exist in this particular one is that this is one we could exist in.

In the long term, gravity may turn out to be a little too strong, and one day it may halt the expansion of the universe and bring it collapsing in upon itself, till it crushes itself down into another singularity, possibly to start the whole process over again. On the other hand it may be too weak and the universe will keep racing away forever until everything is so far apart that there is no chance of material interactions, so that the universe becomes a place that is inert and dead, but very keep racing away. The third option is that gravity is just right − “critical density” is the cosmologists’ term for it − and that it will hold the universe together at just the right dimensions to allow things to go on indefinitely. Cosmologists in their lighter moments sometimes call this the Goldilocks effect − that everything is just right.

You can never get to the edge of the universe. That’s not because it would take too long to get there − though of course it would − but because even if you traveled outward and outward in a straight line, indefinitely and pugnaciously, you would never arrive at an outer boundary. Instead, you would come back to where you began (at which point, presumably, you would rather lose heart in the exercise and give up). The reason for this is that the universe bends, in a way we can’t adequately imagine, in conformance with Einstein’s theory of relativity.

For a long time the Big Bang theory had one gaping hole that troubled a lot of people − namely, that it couldn’t explain how we got here. Although 98 percent of all the matter that exists was created with the Big Bang, that matter consisted exclusively of light gases: the helium, hydrogen, and lithium. Not one particle of the heavy stuff so vital to our own being − carbon, nitrogen, oxygen, and all the rest − emerged from the gaseous brew of creation. But − and here’s the troubling point − to forge these heavy elements, you need the kind of heat and energy of a Big Bang. Yet there has been only one Big Bang and it didn’t produce them.

Date: 2016-04-22; view: 1227