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Dear Dr. Parsons,

I do not wish to be disrespectful, but I do not think Boltzman abstract brains and Hilbert’s abstract hotel have anything to do with how the real Universe is occurring for the following real reason:

Do let me know what you think about this: This is my single unified theorem of how the real Universe is occurring: Newton was wrong about abstract gravity; Einstein...

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Dear Dr. Parsons,

I do not wish to be disrespectful, but I do not think Boltzman abstract brains and Hilbert’s abstract hotel have anything to do with how the real Universe is occurring for the following real reason:

Do let me know what you think about this: This is my single unified theorem of how the real Universe is occurring: Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of abstract NOTHING. Proof exists that every real astronomer looking through a real telescope has failed to notice that each of the real galaxies he has observed is unique as to its structure and its perceived distance from all other real galaxies. Each real star is unique as to its structure and its perceived distance apart from all other real stars. Every real scientist who has peered at real snowflakes through a real microscope has concluded that each real snowflake is unique as to its structure. Real structure is unique, once. Unique, once does not consist of abstract amounts of abstract quanta. Based on one’s normal observation, one must conclude that all of the stars, all of the planets, all of the asteroids, all of the comets, all of the meteors, all of the specks of astral dust and all real objects have only one real thing in common. Each real object has a real material surface that seems to be attached to a material sub-surface. All surfaces, no matter the apparent degree of separation, must travel at the same constant speed. No matter in which direction one looks, one will only ever see a plethora of real surfaces and those surfaces must all be traveling at the same constant speed or else it would be physically impossible for one to observe them instantly and simultaneously. Real surfaces are easy to spot because they are well lighted. Real light does not travel far from its source as can be confirmed by looking at the real stars, or a real lightning bolt. Reflected light needs to adhere to a surface in order for it to be observed, which means that real light cannot have a surface of its own. Real light must be the only stationary substance in the real Universe. The stars remain in place due to astral radiation. The planets orbit because of atmospheric accumulation. There is no space.

Warm regards,

Joe Fisher

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Dear Conrad,

I liked much your essay, I gave it a high rate and I included it in the (second) list of best essays of my review. You seem to gather unanimity here, so I think the interesting question is: who would think otherwise (believe in physical infinity) ? You wrote in your comment that you "started out believing in physical infinity" yourself. Was it just a default position by lack of...

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Dear Conrad,

I liked much your essay, I gave it a high rate and I included it in the (second) list of best essays of my review. You seem to gather unanimity here, so I think the interesting question is: who would think otherwise (believe in physical infinity) ? You wrote in your comment that you "started out believing in physical infinity" yourself. Was it just a default position by lack of precise ideas ? Do you know any physicists having a firm belief in physical infinity ?

If I had to express a bet with respect to cosmology, I would opt for the idea of a spherical universe, that I see as the simplest and most natural way a universe can be created. Indeed, I hardly see the sense and possibility of an infinite universe (how can it start in the first place ?), and I don't believe I have clones anywhere. So, since we already verified the surprising fact that the cosmological constant is extremely small (compared to its microphysical causes) but nonzero, I see it natural to expect a similar property for the curvature of the universal "geography".

But the other question is that of the infinitely small. You seem to assume that nobody takes seriously the idea of a physical infinity in the infinitely small, as all we can measure is approximations. But I do think that there are many people whose views logically imply the existence of a physical infinity in the infinitely small, even if they are not ready to admit it. What I mean here is that they have mutually contradictory beliefs and they fail to notice the contradiction.

Precisely, I see only 3 possibly coherent views with respect to the infinitely small:

1) A digital universe, made of pixels (or the like), where continuous geometrical symmetries are only an emergent property.

2) A quantum universe, where the (usually called "paradoxical") properties of quantum physics are accepted as actually describing how things are, and finally understood as not really paradoxical since they are the solution of this other paradox : the reconciliation of continuous geometrical symmetries with the absence of actual infinity in the infinitely small. This is achieved by the fact that the continuous symmetries (such as rotations of a local object) are not acting over an actually infinite list of really distinct states, but over the continuous values of probabilities for the system to appear in one or another state if it is measured. This means to reject physical realism, as the continuity of the transition between the possibilities for 2 states to be identical or distinct, means that there is no physical reality of which state a system exactly is in. I commented this further in pages 5 and 6 of my essay.

3) A classical continuous universe, which logically means to admit an actual infinity of physically distinct possible intermediate states between 2 states. A typical example is Bohmian mechanics. Supporters of such views may hope to keep this compatible with practical finiteness, i.e. that this actual infinity only concerns the ontology that, at the same time, they wish to deny on an effective level, where it would behave as a potential infinity only. Namely, they expect the effects of the whole infinity of decimals of their "hidden variables" are not popping up in finite times. However I do not see it clear if they can really find a coherent theory satisfying that property and that would be compatible with known physics (quantum field theory). For details, see in my criticism of Bohmian mechanics, the section "Problem 2 : the nonsense of deterministic randomness".

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Dear Bill,

What a delightful essay! You should consider moonlighting as a science writer (what is a physicist-in-residence, anyway?)

"..I reject physical infinity, for three reasons. First, mathematically, it

makes computations intractable. Second, operationally, I do not know how—even in principle—how

to observe, measure or manipulate physically infinite objects...

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Dear Bill,

What a delightful essay! You should consider moonlighting as a science writer (what is a physicist-in-residence, anyway?)

"..I reject physical infinity, for three reasons. First, mathematically, it

makes computations intractable. Second, operationally, I do not know how—even in principle—how

to observe, measure or manipulate physically infinite objects or systems. Third, conceptually, it

embodies a viciously unphysical ontology, namely, that physical constituent parts can equal each

other and the physical whole from which they derive."

These are all good reasons, but may I suggest that infinities in physical theories may have a useful role to play that is in my opinion still greatly under-appreciated: I think that at least in some (perhaps, with enough imagination, in all meaningful) cases in which they occur, they may be telling us that we are not looking at the physical situation at hand in "the right way".

The paradigm example to me is the Lorentz factor. For v=c it is infinite, and so presumably one of the unfortunate victims of your effort to eradicate its kin from physics. But what if we look at its inverse: The inverse of the Lorentz Factor tells us how much the proper time changes with respect to coordinate time. In fact, because of the mathematical form of gamma we can get it to tell us more: How much of the proper time is "projected" unto coordinate time (as I'm sure you know, one can easily see this by drawing the appropriate triangle that illustrates



In that case, if we take the triangle relationship seriously, gamma=1 tells us that all of the object's proper time is "projected" unto the observer's coordinate time and gamma=infinity tells us that none of it is "projected" unto the observer's coordinate time, or, in other words, that the object's proper time is orthogonal to the coordinate time if we were to assign unit vectors to the abstract plane spanned by the two time parameters . This is of course consistent with the fact that null vectors are orthogonal to time-like vectors.

Orthogonality is one of those situations which commonly involves zero and infinity, and seems to have been what lurked behind this infinity. Orthogonality is also a basic conceptual staple of physics, and so I suspect that there is something conceptually very clear and thoroughly physical behind many infinities in physics in a similar manner, but not very well recognized as such.

I'd be interested to know what you think of this argument, and whether it leads you to modify your categorical rejection of infinities in physics.

Best wishes,

Armin

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Dear William T. Parsons,

I like the topic and the style of your essay and I agree with your main stance (mathematical infinity is fine, physical infinity is something else). However, I do think there are a few flaws as well.

One minor thing is that you state that the lemniscate is the common and correct symbol for mathematical infinity. You use it both for infinity in the sense...

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Dear William T. Parsons,

I like the topic and the style of your essay and I agree with your main stance (mathematical infinity is fine, physical infinity is something else). However, I do think there are a few flaws as well.

One minor thing is that you state that the lemniscate is the common and correct symbol for mathematical infinity. You use it both for infinity in the sense of calculus (you mention limit and series) and set theory. However, I don't think the use of the lemniscate symbol is appropiate in the latter case, or in the section on Hilbert's paradox of the Grand Hotel, which is about countably infinite sets. For infinite sets it is possible to rigorously define operations on the infinite cardinalities or ordinals. This is not "improper". (Later on, you mention lemniscate + lemniscate among examples of meaningless use of infinity, but this seems fine, even within calculus; the example makes more sense with a minus sign, which does lead to an indeterminacy within the context of calculus.)

There are some sloppy phrasings: you write "subsets of infinity", rather than "subsets of infinite sets" and you state Dedekind's definition of infinity in terms of equality of a set with a subset, but this has to be equality _of size_ of those sets. I agree that the standard of rigour can be a bit lower in these kind of essays, but in these examples, it requires little effort (and no loss of accessibility) to make it more accurate.

You write yourself that "the mathematics of infinity is tricky". I would like to add: especially if you mix it with probabilities! Indeed, my main complaint is about the probabilistic argument on p. 7: as it is written here, you suggest that it is possible to use (infinite) cardinalities as a basis for a probability measure, but such a measure simply isn't defined. There are ways to define probability measures in such sitations, for instance using natural density. (Or using non-standard measures, something I work on, but which is not even needed for the case at hand.) On none of the approaches I known of, it will follow that all subsets of the same cardinality have the same probability measure. For instance, the natural density of the set of even natural number is 1/2, although it has the same cardinality as the entire set of natural numbers (namely, countably infinite): this is easy to interpret as the probability that a random natural number is even. And this better be the case, otherwise you come into conflict with the basic (finite) additivity axiom of probability theory. (Consider the subsets of even and odd numbers to see this point.) In other words, I don't buy your 50-50 argument for the Boltzmann brains vs Universe odds. ;-) (I can send you references if you would be interested in the details of mixing infinity and probability.)

I do like your use of the Bravo card (actually, this idea is quite close to that of a non-standard natural number, which is an alternative way of dealing with infinity in the context of calculus) and I do think most of this text would be enthusing for a lay audience, so my vote is a 7/10.

Best wishes,

Sylvia Wenmackers - Essay Children of the Cosmos

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