Part One.

The Finite Universe

Empirically, the universe in which we live is observed to be finite. It only goes so far, in any direction you care to point. The limits to our observational abilities are now knowable, although a century ago they were not. We have tools of thought at our disposal which were not imaginable in previous eras, and we are getting pretty good at knowing what we can know. The field of knowledge I am bringing to bear in this section is astronomy, with peripheral reference to astrophysics and cosmology. We cannot see the "far edges" of the universe, but we can see enough of the universe to know that it isn't infinite. The primary perception adduced in evidence is universal recessional red shift. Everything out there in distant galaxies is leaving us in a hurry.

This observation resolves the historical dilemma known as Olber's Paradox. If there were an infinite number of stars, shining for an infinitely long time, their light would add up. It would fill the sky and pile up on us, toasting us to a crisp. We know now the resolution to this. There are not an infinite number of stars, and not one of them has been shining for an infinitely long time. They're all flying away from us, and the farther they get the faster they flee. (Mentioning paradox, I feel that introducing infinity into a logic system intrinsically opens the logic system to paradox; note that most of the classical paradoxes, before Bertrand Russell, involved infinity explicitly or implicitly.)

What we can see now is the Big Bang picture: since the galaxies are scattering, formerly they were closer, so it seems like there was a time when they were pretty much all together. The theoretical model gets much more complex, but all we need is the roughest shape of it to establish the critical point: there wasn't any forever. There wasn't any eternity. There was a time the universe started, and we can put brackets around when that was: it must have been between this time and that time.

Immediately we see that eternity is out the window. There cannot be an infinite duration stretching forward and backward in time, for we have just set a limit on past time. Time started when the universe started, and that's it. If your mind insists on wiggling around a fact, to try to take exception or to dislodge or negate that fact, that's a malfunction in your mind rather than something wrong with the fact. You may take comfort that it is a malfunction common to humans. However, it cannot be considered commendable nor particularly useful.

It seems natural to wonder what happened before time started, but you must realize those thoughts will not teach you any more about reality, just maybe about the inside of your own mind. Meditation is a utility which is only helpful as long as you can learn from it. If you can't learn, well, at least it's keeping you out of other people's way, but for you it's not so good, because it's uselessly, wastefully burning up some of your precious moments of life. You don't have forever, believe it or not.

We can't see all the galaxies, but we don't need to, to trace their trajectories back in time, in order to see that they were once together. All we need is to see a few, maybe a dozen of them, to establish the fact of the Big Bang. Nature in Her generosity has provided us with several million times more galaxies than we needed to learn this fact.

Space isn't there if it's empty. That's another notion we need to squeeze into our squirming minds. One can form a concept of an immeasurable vacuum, with neither matter nor light. The analysis must be that if it's got nothing in it, it's literally immeasurable, in other words it isn't there. Nature ignores a vacuum. Truly empty space has got no metric: if you can't put a yardstick in it to measure it, or (cheaper) shoot a photon through it to measure it, there is no way to distinguish it from imaginary space, or any other delusion. Space in nature, the real outer space of the universe, is chock full of photons zipping through it. It can be measured, and it is not empty. Time passes in it, it is warped by gravity, and it may contain matter, maybe like cosmic rays zipping through it, or the occasional lonely proton. It is not physically meaningful to postulate space without mass nor energy.

We see that the Big Bang event which started time, and gave us mass and energy, and cranked up gravity, also exuded space. It wasn't around before. Space is existentially dependent on mass and energy, just like time is. It is important to keep in mind that the Big Bang was a finite event. It was characterized by a specific energy level, which resulted in a certain amount of mass being produced, generating a limited amount of space over a particular period of time. That is our universe, and we are its creatures. There were no infinities in the catalog. Infinities are our creations, one and all, and they are symptoms of our mental disorder. They have no objective existence outside of our warped collective imagination.

Stretch out your arm and point your finger. Space goes that way a long way. It doesn't go on forever. It extends that way as far as the outermost galaxies have had a chance to get since the Big Bang, and as far beyond that as their light has had a chance to get, since the stars in those galaxies have been shining. That's a long way. But it isn't forever, and there's a big difference. It's a certain number of kilometers, though it's changed and gotten bigger, lots more kilometers while your finger has been pointing. You can put your arm down now. I don't know that number, of how many kilometers space stretches that way. You don't either. Like I said, space expanded while you were pointing. It's bigger now, but it's not infinite. It never will be infinite, and never could be. Infinity does not exist in nature.

Part Two.

The Gravitational Monobloc

The existence of space/time logically precludes infinitude of mass/energy.

If you allow that gravity is an inevitable concomitant of mass, there can be no infinitude of mass in any conceivable universe, for an infinitude of gravity would disallow extension. Space and time could not coexist with infinite gravity. A universe consisting of a singularity with no extension may not be distinguished in any way from a non-existent universe, so we are safe in the general statement there can be no infinite mass.

There ia a parallel argument for time; time is derived from motion, only possible when space is extended. Time is derived from motion and is not a meaningful metric without the concomitant of space. Time without events collapses into the equivalent statement no-time. So since time exists there is no infinite mass, no infinite gravity in our own universe, nor any other universe which can exist.

The postulate of spacetime without massenergy is nonphysical and collapses into a semantically equivalent statement of nonexistence, for lack of a supporting metric.

The energy arguments are similar to the above results for infinite mass, for infinite energy must have zero wavelength, which also disallows extension. Since (by special relativity) energy and mass are interconvertible, the gravitational argument likewise applies to infinite energy. Infinite energy is indistinguishable from the case of infinite mass; neither would allow existence, and so these are precluded by existence. There is some space and some time; in no universe which has any space and any time can there be an infinity of mass or energy. We have shown that material infinities are strictly impossible.

The monobloc of Parmenides, in which time, motion and change are all illusory, is strikingly similar. That was the most dismal of the pre-Socratic speculations. It is hard to imagine how anyone could ever think up anything so boring.

Part Three.

Mathematics and its Methodology

Mathematicians continually bewail efforts to induce them to use instrumentalist approaches to their dicipline, complaining that it clutters up their elegant theoretical constructs with ugly physical reality. They have failed to take notice that they are left alone in their Platonic Realm of Ideals, the other branches of knowledge having abandoned that room a couple centuries ago for some very good reasons. Work performed within the Realm of Ideals has two outstanding flaws: first, it doesn't exist, and second, it is meaningless. Otherwise, no doubt it can be quite pretty, to someone schooled to its understanding.

The constructivist outlook is that one who would presume to prove anything, be constrained to take account of some fairly recent realizations. First, no open-ended assignments can be given any truth value whatsoever. Without cloture, there is no operation. This means, for example, that you may not use an endless iteration of any kind in any circumstance and say, that were it ever finished, it would have a particular implication. It would not, in fact, have the implication you meant, because no statement involving an endless iteration can have any meaning whatsoever. You may not tell a student to divide a number line in half, repeat, then do it forever. That is sheer lunacy.

To a much greater degree, since the popularization of computers, logicians have become highly aware of the energetic costs of performing logical and numerical calculations. Such features of the computer world as storage requirements and running-time estimates have become abstract enough, while remaining sharply predictable, that they may now be fed back into the parent diciplines of logic and mathematics, and be used to increase the efficiency and productivity of these more general studies. Every mathematical operation must be recognized to take a finite time. Proofs must fulfill the criterion that the equation corresponding to their run time must be demonstrably solvable, or in other words, for a proof to be valid it must be able to be worked in finite time, as well as fulfill the other, more traditional forms of internal validity.

From this new perspective, we may see that many of the old familiar mathematical techniques, and many of the old proofs, would no longer be acceptable with these guidelines, for so many of them are based on dubious techniques such as obtaining asymtotes by endless iteration. Any proof containing an open-ended procedure, or endless iteration, would fail to give assurance of cloture in the run-time asessment, so regardless of its plausibility would not be considered a valid proof. This criterion, or its functional equivalent, could help reduce the number of frivolous or meaningless mathematical theorems for which proof is claimed. It makes the concept of proof more meaningful, as something which can be performed in a finite time with a finite number of steps. Proofs which satisfy the finite run-time criterion may be easily be seen to be stronger than those which do not. The category of proofs which cannot be shown as having a finite run time, whether by needing an endless sequence of operations or by logical structure which makes their run time indeterminable, in other words failing to guarantee completion, may be rejected as not meaningful.

Part Four.

The Number Line

No number line which has a one-to-one correspondence with any aspect of existence contains an infinite number of points. Again. No number line which has a one-to-one correspondence with reality has an infinite number of points. To reiterate. No number line which has a one-to-one correspondence with any meaningful facet of the material universe has an infinite number of points. Once more with feeling. No number line which has anything to do with physics can contain an infinite number of points.

This is just for the record, to corrrect an unfortunate error. It has to do with an issue of perspective. Mathematics has been viewed as the perfect template, to which reality sadly fails to measure up. This view is but an unconsious echo of monotheist bias. It is time to shift this perspective to one based on reality. From this point of view, we may see that mathematics as we now have it is just a lousy tool. It doesn't fit very well with anything we observe, because it always skews off at a tangent to follow some illusion.

Now we can see what to do. Clearly, moaning about why the real fails to fit with our hallucination is so much wasted energy. What has to be done is to repair the tool. The map should be adjusted until it corresponds to the territory, because you can't change the territory to fit the map. Among the very first priorities in reforming mathematics should be throwing away all infinities. There aren't any such things, and they're impossible. They don't even fit in with the rest of the objects in the collection. Infinity is not a quantity, and infinity is not a quality. Infinity is not a subject, and infinity is not an object. (There is a distinct category in which infinity can be included, which we have within the Pagan community. Infinity is a spell.)

Take your number line. Put a tack in it where infinity should be included. Ah, so it doesn't fit on there anywhere? That should tell you that infinity is not a number. The reason it's not like the other quantities is that infinity is not a quantity. It's not a particular quantity, and it's not even a general quantity. It's not a specific number, and it's not a number collection. It's a ringer, a fake, phony, fraud, forgery, counterfeit, a disguised demon. The most fundamental rules of arithmetic do not include it, and fundamental arithmetic has got to be warped and twisted to allow for its inclusion. Infinity cannnot be generated by applying simple operations to the integers an integral number of times. Numbers can be produced, but not infinity. Infinity isn't a number. It isn't anything.

Stretch a number line out as far as you want to. Stretch it out all across the Galaxy. If that's too cramped for you, stretch it out from the Milky Way to the Large Magellanic Cloud, a neighbor. Chop up that number line as many times as you want to. We know that every operation takes time, so you can have a bunch of helpers to help you chop in parallel. These are fault-free number-line-chopping machines which can make a bunch of number-line chops every second, so you can finish your chopping in a reasonable time. You cannot chop any stated-length number line into an infinite number of points. Even though you reach completion on the operation, there is not an infinite number of points available on a physical number line.

The metric of space is not a continuum, but is granular. Mass is not a continuity on the small scale, but is composed of atoms. Energy is not continuous on the small scale, but is composed of photons. Time is not continuous on the small scale, but is composed of chronons. Gravity is not continuous on the small scale, but is composed of gravitons. Space is not continuous on the small scale, but is composed of metric quanta called the Planck length. No smaller unit of measurement is physically meaningful. The Planck length is the quantum of extension, the basic granularity of space. A physical number line is composed of a finite number of these quanta. It has that many points on it. This is the nature of physical reality. Should you prefer to live in a universe which is continuous rather than granular, a universe in which quantum physics does not apply, you are free to dwell in your delusion, and arrange your mathematics on that basis. You are not free to impose your preferred delusion on the remainder of humanity.