DOING AWAY WITH INFINITY SOLVES MUCH MATH & PHYSICS

Particle physics: Fundamental physics is frustrating physicists: No GUTs, no glory, intones the Economist, January 11, 2018. Is this partly caused by fundamental flaws in logic? That’s what I long suggested.

Says The Economist:“Persistence in the face of adversity is a virtue… physicists have been nothing if not persistent. Yet it is an uncomfortable fact that the relentless pursuit of ever bigger and better experiments in their field is driven as much by belief as by evidence. The core of this belief is that Nature’s rules should be mathematically elegant. So far, they have been, so it is not a belief without foundation. But the conviction that the truth must be mathematically elegant can easily lead to a false obverse: that what is mathematically elegant must be true. Hence the unwillingness to give up on GUTs and supersymmetry.”

Mathematical elegance? Define mathematics, define elegance. What is mathematics already? What maybe at fault is the logic, that is the mathematics, brought to bear in present day theoretical physics. And I will say even more: all of today logic may be at fault (what logic is, is itself the deepest problem in logic…). It’s not just physics which should tremble. The Economist gives a good description of the developing situation, arguably the greatest standstill in physics in four centuries:

“In the dark

GUTs are among several long-established theories that remain stubbornly unsupported by the big, costly experiments testing them. Supersymmetry, which posits that all known fundamental particles have a heavier supersymmetric partner, called a sparticle, is another creature of the seventies that remains in limbo. ADD, a relative newcomer (it is barely 20 years old), proposes the existence of extra dimensions beyond the familiar four: the three of space and the one of time. These other dimensions, if they exist, remain hidden from those searching for them.

Finally, theories that touch on the composition of dark matter (of which supersymmetry is one, but not the only one) have also suffered blows in the past few years. The existence of this mysterious stuff, which is thought to make up almost 85% of the matter in the universe, can be inferred from its gravitational effects on the motion of galaxies. Yet no experiment has glimpsed any of the menagerie of hypothetical particles physicists have speculated might compose it.

Despite the dearth of data, the answers that all these theories offer to some of the most vexing questions in physics are so elegant that they populate postgraduate textbooks. As Peter Woit of Columbia University observes, “Over time, these ideas became institutionalised. People stopped thinking of them as speculative.” That is understandable, for they appear to have great explanatory power.“

A lot of the theories found in theoretical physics “go to infinity”, and a lot of their properties depend upon infinity computations (for example “renormalization”). Also a lot of problems which appear and that, say, “supersymmetry” tries to “solve”, have to do with turning around infinite computations which go mad for all to see. For example, plethora of virtual particles make Quantum Field Theory… and miss reality by a factor of 10^120 (one followed by 120 zeroes…). Thus curiously, Quantum Field Theory is both the most precise, and most false theory ever devised. Confronted to all this, physicists have tried to do what has NOT worked in the past, sometimes for centuries, like finding the intellectual keys below the same lighted lamp post, and counting the same angels on the same pinhead.

A radical way out presents itself to simplify the situation. It is itself very simple. And it is global, clearing out much of logic, mathematics and physics, of a dreadful madness which has seized those fields: GETTING RID OF INFINITY… at the logical level. Observe that infinity itself is not just a mathematical hypothesis, it is a mathematically impossible hypothesis: infinity is not an object. Infinity has been used as a device (for computations in mathematics). But what if that device is not an object, is not constructible?

Then lots of the problems theoretical physics try to solve, a lot of these “infinities“, simply disappear. 

Colliding Galaxies In the X Ray Spectrum (Spitzer Telescope, NASA). Very very very big is not infinity! We have no time for infinity!

A conventional way to get rid of infinities in physics is to cancel particles with particles: “as a Higgs boson moves through space, it encounters “virtual” versions of Standard Model particles (like photons and electrons) that are constantly popping in and out of existence. According to the Standard Model, these interactions drive the mass of the Higgs up to improbable values. In supersymmetry, however, they are cancelled out by interactions with their sparticle equivalents.” Having a finite cut-off would do the same.

A related logic creates the difficulty with Dark Matter, in my opinion. Here is why. Usual Quantum Mechanics assumes the existence of infinity in the basic formalism of Quantum Mechanics. This brings the non-prediction of Dark Matter. Some physicists will scoff: infinity? In Quantum Mechanics? However, the Hilbert spaces which Quantum Mechanical formalism uses are often infinite in extent. Crucial to Quantum Mechanics formalism, but still extraneous to it, festers an ubiquitous instantaneous collapse (semantically partly erased as “decoherence” nowadays). “Instantaneous” is the inverse of “infinity” (in perverse infinity logic). If the later has got to go, so does the former. As it is Quantum Mechanics depends upon infinity. Removing the latter requires us to change the former.

Laplace did exactly this with gravity around 1800 CE. Laplace removed the infinity in gravitation, which had aggravated Isaac Newton, a century earlier. Laplace made gravity into a field theory, with gravitation propagating at finite speed, and thus predicted gravitational waves (relativized by Poincaré in 1905).

Thus, doing away with infinity makes GUTS’ logic faulty, and predicts Dark Matter, and even Dark Energy, in one blow.

If one new hypothesis puts in a new light, and explains, much of physics in one blow, it has got to be right.

Besides doing away with infinity would clean out a lot of hopelessly all-too-sophisticated mathematics, which shouldn’t even exist, IMHO. By the way, computers don’t use infinity (as I said, infinity can’t be defined, let alone constructed).

Sometimes one has to let go of the past, drastically. Theories of infinity should go the way of those crystal balls theories which were supposed to explain the universe: silly stuff, collective madness.

Patrice Aymé

Notes: What do I mean by infinity not constructible? There are two approaches to mathematics:1) counting on one’s digits, out of which comes all of arithmetics. If one counts on one’s digits, one runs of digits after a while, as any computer knows, and I have made into a global objection, by observing that, de facto, there is a largest number (contrarily to what fake, yet time-honored, 25 centuries old “proofs” pretend to demonstrate; basically the “proof” assumes what it pretends to demonstrate, by claiming that, once one has “N”, there is always “N + 1”).

2) Set theory. Set theory is about sets. An example of “set” could be the set of all atoms in the universe. That may, or may not, be “infinite”. In any case, it is not “constructible”, not even to be extended consideration, precisely because it is so considerable (conventional Special Relativity, let alone basic practicality prevent that; Axiomatic Set Theory a la Bertrand Russell has tried to turn around infinity with the notion of  a proper class…)

In both 1) and 2), infinite can’t be considered, precisely, because it doesn’t finish.

Some will scoff, that I am going back to Zeno’s paradox, being baffled by what baffled Zeno. But I know Zeno, he is a friend of mine. My own theory explains Zeno’s paradox. And, in any case, so does Cauchy’s theory of limits (which depends upon infinity only superficially; even infinitesimal theory, aka non-standard analysis, from Leibnitz + Model Theory survives my scathing refounding of all of logics, math, physics).  

By the way, this is all so true that mathematicians have developed still another notion, which makes, de facto, logic local, and spurn infinity, namely Category Theory. Category Theory is very practical, but also an implicit admission that mathematicians don’t need infinity to make mathematics. Category Theory has now become fashionable in some corners of theoretical physics.

3) The famous mathematician Brouwer threw out some of the famous mathematical results he had himself established, on grounds somewhat similar to those evoked above, when he promoted “Intuitionism”. The latter field was started by Émile Borel and Henri Lebesgue (of the Lebesgue integral), two important French analysts, viewed as  semi-intuitionists. They elaborated a constructive treatment of the continuum (the real line, R), leading to the definition of the Borel hierarchy. For Borel and Lebesgue considering the set of all sets of real numbers is meaningless, and therefore has to be replaced by a hierarchy of subsets that do have a clear description. My own position is much more radical, and can be described as ultra-finitism: it does away even with so-called “potential infinity” (this is how I get rid of many infinities in physics, which truly are artefacts from mathematical infinity).  I expect no sympathy: thousands of mathematicians live off infinity.

4) Let me help those who want to cling to infinity. I would propose two sort of mathematical problems: 1) those who can be solved when considered in Ultra Finite mathematics  (“UF”). 2) Those which stay hard, not yet solved, even in UF mathematics.

Tags: GUT, Infinity, Theoretical Physics