The finite and the infinite

A review of “The beginning of infinity” by David Deutsch, and other thoughts Jonathan Allin This is partly a review of “The beginning of infinity” by David Deutsch, and partly reflections on the finite and infinite prompted by the book. I’ve attempted to bring together ideas from philosophy, science and cosmology, mathematics, and of course Jewish thinking.

My excuse for including this review is that Deutsch, a theoretical physicist, is Jewish, was born in Haifa, educated in Cambridge and the Other Place, and now works at the Other Place.

A better title for Deutsch’s book would be “The beginning of universality”. Why do some ideas, such as the alphabet, writing, commerce, scientific ideas, become ubiquitous? Unfortunately Deutsch deliberately conflates infinity with “unbounded” and with universality, treating these three separate concepts (to my mind) as a single concept. Deutsch makes a valid point: a good scientific idea has survived debate and criticism, and will thus over time become universal. Indeed this must be the definition of good science, which nonetheless doesn’t preclude ideas from evolving or being replaced with better ideas that fit more of the available observations. The history of philosophy, on the other hand, is at the mercy of inevitably sketchy, inaccurate, or ambiguous record keeping. We don’t really know what Socrates said, but only what Plato reported that he said. Whereas science has unambiguously moved forward, the same cannot be said of our social or political systems: we continue to suffer decades of misery interspersed with short periods of social enlightenment. Deutsch states that human progress started with the Enlightenment (from the end of the 17th century to the start of the 19th century, the “long” century), and even then distinguishes between the British Enlightenment and the Continental Enlightenment. The former was

about gradual improvement towards an ideal, the latter about achieving a Utopia directly, an idea that lead to the bloodshed of the French revolution. I find this hard to accept: humans have been around for less than a million years (compared to the dinosaurs who were around from 230 million years ago to 65 million years ago), civilisation for less than 10,000 years, and the Enlightenment for a few hundred years. How long can we seriously expect the growth of science and knowledge to continue? The premise behind Deutsch’s book is that humans have the distinction of being universal explainers and constructors, that is to say that we have the ability to explain everything and build anything. Given time and energy there are no limits on what we can achieve: we could bend whole galaxies to our ends. From this he suggests that all sentient beings throughout the universe, having become universal explainers and constructors, would be on a par. Computing power compensates for limited memory and computational ability, and so brings all to the same level. However I’m not sure this is the whole story: whilst all sentient beings may become universal constructors, would they have the same self-awareness or level of consciousness?

At the simplest level we are unique in our story telling, stories and legends that were our attempt to explain the world and which became religious beliefs. But these cannot be good explanations: to say that crops fail or winter comes because the gods are angry or have deserted their worshippers does not offer any useful insight that can be built upon, whereas the scientific method does.

Unfortunately Deutsch builds his book on concepts which he fails to define. “Explanation” is one of these. Science does not, and cannot, “explain” anything, in the sense that it cannot answer the question “why”. All science can do is look for canonical, and ultimately approximate, relationships between observed events. For example Newton’s equations of motion and gravity brought together many disparate observations from Galileo, Kepler, and others, whilst debunking Aristotle’s belief that rest is the natural state of motion to which all things return. However Newton didn’t and couldn’t explain

time or space. Einstein’s theories of special and general relativity built on the works of Newton, as well as Galileo, Mach, Maxwell and others. He was able to incorporate in his field equations the hitherto anomalous precession of Mercury’s orbit, the constancy of the speed of light and other massless particles, and other relativistic effects. However Einstein doesn’t have the last word. We are yet to reconcile relativity with quantum physics, that other bastion of 20th century science.

I suspect that Deutsch picked up some Talmudic training, given his interchangeable use of “infinite” and “unbounded”. The latter is perhaps closer to the Hebrew: וףֹס אין )without end) or גבול בלי (without limits). The Greek for infinity (ἄπειρον) can also mean unlimited, though can also mean chaotic. In Greek thinking infinity had negative connotations: in Aristotle's words, "... being infinite is a privation, not a perfection but the absence of a limit...". Pythagoras believed that any given aspect of the world could be represented by a finite arrangement of natural numbers (that is, the integers). Plato believed that even his ultimate form, the Good, must be finite and definite.

There is an interesting association between Tarot and Kabbalah. There are 22 Tarot cards in the Major Arcana (greater secrets), which corresponds nicely with the number of letters in the Hebrew alphabet. In the 1700s the infinity symbol (∞) began to appear on the Tarot card known as the Juggler or the Magus, which has the associated letter ℵ. Georg Cantor was the founder of the modern mathematical theory of the infinite and of set theory. He was a devout Lutheran (though almost certainly of Jewish descent) which strongly influenced his nondeterministic views. Cantor used the symbol ℵ 0 to stand for the first infinite number, though probably not because of his Jewish roots.

Mathematicians have a history of abusing the concept of infinity. David Hilbert proposes a hotel with infinitely many rooms, his Hilbert Hotel. A traveller arrives wanting a room, but every room is occupied. “No problem”, says the desk clerk, “I’ll just ask each guest to move into the

next numbered room and you can have room number 1”. Of course contacting an infinite number of guests would take infinite time and energy, which is rather a big problem.

Bertrand Russell was the first to publish the paradox in Georg Cantor’s set theory (Russell’s paradox): “Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves”. The resolution to this paradox lead to “first order logic” (which is the foundation for all deductive logic) and in particular to Zermelo–Fraenkel set theory. The price paid is that first order logic is incapable of describing a system with an infinite domain. But is it meaningful to talk of a system with an infinite domain?

I’m going to claim that most, if not all, of the well-known paradoxes in mathematics, are the result of abusing infinity.

It’s surprisingly hard to find a mathematical definition of infinity.

This idea of infinity could be interpreted as “for any number you can give me, I can give you back a bigger number”. The useful aspect of thinking of infinity as meaning “unlimited” in this way, is that any number we generate can be manipulated (added, multiplied, compared, and so on). However, in the tradition of Alice’s Humpty Dumpty I shall use this idea to define “unlimited”, rather than “infinite”. As an example, Euclid’s proof that there is no largest prime number relies on this idea of “unlimited”, as does any proof by induction.

Or we could define infinity as 1 divided by 0. This definition generally gives the arithmetical results we might expect. For instance, using this definition, we can see that infinity plus infinity gives infinity, or any number times infinity (including infinity times infinity) gives infinity. Importantly with this definition, zero times infinity, and infinity divided by infinity, is undefined.

This brings us to a strange aspect of infinity. If we use our definition of “unlimited”, then numbers behave properly. Given any suitably large number, we can say that half will be even, one third will be divisible by three, one fifth divisible by five, and so on. However when we talk about an infinity of numbers, an infinity of them will be even numbers, an infinity will be divisible by three, an infinity divisible by five, and so on. But because we can’t compare infinities (that is, we can’t say whether one countable infinity is bigger than another) it’s meaningless to ask what percentage of an infinite set of numbers are, say, divisible by five. As another example, we might say that planets suitable for carbon-based life are outrageously rare, but if the universe is infinite we can no longer talk about the likelihood of a planet hosting life. Deutsch attempts to go around this problem by the use of “measures”. Unfortunately his argument is circular.

Zeno’s claim that Achilles can never catch the tortoise is a well-known apparent paradox. However it isn’t really a paradox because attempting to add an infinite number of infinitesimals will always be problematic. Another way of stating Zeno’s paradox is to ask “what is the sum of 1/2 + 1/4 + 1/8, and so on to infinity?” You may guess that the answer is 1. However this is really a definition of the sequence, rather than an independently assertible fact: limit theory only shows that for any small value we choose, we can add enough terms such that difference between the sum and 1 will be less than this value. Which is closer to the idea of an “unlimited” number of terms (our first definition), rather than an infinite number of terms (our second definition).

There are a couple of takeaways from this somewhat esoteric discussion:

If an infinity crops up in a mathematical expression, all bets are off. Infinities cannot be normalised. Deutsch’s statement that “mathematicians realised a long time ago that it’s possible to work with infinity” isn’t tenable

We are not attempting to say what infinity “is”, but only to provide a definition we can work with

Infinity, both in its every day use, and its use in mathematics and cosmology, is a concept not a number

Perhaps one aspect of God that all will agree on, whether or not we accept such an existence, is that God must be infinite. Judaism generally regards the Universe as finite, however there is no non-Divine process that can bridge the gap between the finite and infinite. For instance to change an object’s speed instantaneously would require infinite power. The 16th century Kabbalist Arizal put forward the idea of צמצום ,or “contraction”, used by an infinite God to create and interact with a finite world. There is a huge philosophy associated with צמצום ,however perhaps all that matters is that all should be able to agree that צמצום is the necessary miracle of the Divine (again, whether one does or does not accept the Divine).

There is a third aspect of God we might all agree with. If we assume a finite universe, we can calculate the number of particles it contains, which is around 10 88 (that is, 1 followed by 88 zeroes). This is a huge number, mainly made up of photons and neutrinos (the stuff of matter, protons, electrons, and so on, that is the “hard stuff”, comprises only 0.1% of the particle count). However, as humongously big as this number is, an infinite God can still devote an infinite amount of attention to every single one of these particles (remember, infinity divided by any number is still infinity).

In general we use comparative descriptions to describe our world. If we say a building is large, we probably mean it’s larger than the average building. This sort of language cannot work when attempting to discuss the infinite. Without boundaries we have nothing to hold onto: we perceive a shape because of the contrast between shape and background. What then, of an infinite God? If we can’t talk about ratios and proportions of an infinite quantity, can God have form? I have a recollection that Maimonides regarded God as formless, and דברים

2:15-18 may support this, “...for you did not see any likeness (תמונה ( on this day...” I believe it was Fred Hoyle who said something like “The universe is either finite or infinite. Neither proposition is plausible”. We should agree on what we mean by “the universe”. In keeping with Humpty Dumpty, by (my) definition there is only one universe. When cosmologists such as Brian Greene talk about a multiverse (perhaps in an attempt to explain quantum indeterminacy or the “Goldilocks” dilemma), it’s less error prone if we think in terms of a single universe that is split into regions which are more or less causally connected.

In an infinite universe stars and galaxies would still be randomly distributed, so each point in the universe is unique (for various reasons the idea that an infinite universe would contain an infinite number of copies of each of us doesn’t hold water). However, as we zoom out to larger scales this fine structure of stars, galaxies, and galactic clusters, would be less and less visible and increasingly irrelevant; at the same time it might be that larger structures start to become apparent. But as we go all the way to an infinite universe, to have a form requires that the universe is made up from a finite number of “large scale” structures, each of which must be of infinite size. I don’t think this tenable.

Perhaps the preceding argument suggests also that an infinite God would be formless. The Oneness of the Divine is the central tenant of Judaism, which does seem to require that God is formless and without large scale structure.

Recent thinking is leaning toward an infinite universe, however this appears to be based on faulty assumptions about the “shape” of the universe. If you’ve read any book on cosmology, it will probably state something like: “the universe either has positive, zero, or negative curvature, leading to a closed, stable, or infinite universe”. For many reasons this description fails. Not only does it fail in two spatial dimensions, it becomes meaningless when we consider 4 dimensions

(three of space and one of time). So even though astronomical measurements may suggest that the expansion of the universe is accelerating, this does not mean that the universe is infinite: it can still be finite and closed.

What is a “closed” universe? A universe could be closed in one or more spatial dimensions, which means that if I set off on a long journey (billions of light years), I will eventually come back to my starting point. Many cosmology books will use the surface of the earth as an analogy: if I set off in a straight line, after about 40,000km I’ll be back where I started. Regrettably this analogy generates more heat than insight. You might also ask, why can’t the universe just come to an end? However I don’t think anyone would propose this as a suitable model because you could then legitimately ask “what is beyond the end?” Deutsch attempts to explain quantum indeterminacy (for instance, when and in what direction will a radioactive nucleus decay) by an “uncountable infinity” of universes. Now an uncountable infinity is yet another idea of infinity, which is an infinity of the continuum. Whereas the integers (1, 2, 3, and so on) are countable even if by some definition they are infinite, the irrational numbers (those with a countably infinite number of decimal digits) are not countable. Cantor desperately attempted to prove that ℵ 1 , which is equivalent to 2 ℵ0 , was equal to C, the infinity of the continuum. This is known at the Continuum Hypothesis.

Kurt Gödel picked up Cantor’s mantle and helped to prove many ideas of set theory and transfinite numbers. Gödel attempted, but failed, to prove the Continuum Hypothesis (and the associated Axiom of Choice). However, what he did show (with a proof completed by Paul Cohen in 1963) was that anything that could be proved for transfinite numbers would have no impact on finite numbers: the finite and transfinite are two separate domains.

Put it this way, explaining our world by invoking an infinite Prime Cause feels no less rational than invoking an uncountable infinity of universes.

My personal Belief (note the capital “B”) is that the universe is spatially closed and finite. That is, the universe has a defined and measurable size.

Maimonides believed the universe to be finite in space (though he also adopted Aristotle’s idea of concentric heavenly spheres with Earth at their centre). Whereas Aristotle believed that the world had existed from eternity, because every event must have a cause, Maimonides believed the eternity of the world was unproveable. God as the Prime Cause, perhaps naively, suggests that the world has a finite history.

What might Aristotle and Maimonides have made of quantum randomness? Would they still be cause-and-effect determinists?

Modern cosmology (which perhaps started in the 1965 with the discovery of the cosmic microwave background) provides strong evidence for a universe that is 13.7 billion years old. Steady state models with an infinite past and future, such as Hoyle’s or Einstein’s, have been superseded. The other option is that the universe is closed in time, which is to say that if I hang around long enough everything that has happened will happen again. Whilst I can get my head around a spatially closed universe, a temporally closed one is beyond me. This leads however to the aforementioned problem: if the universe is finite in time but not closed, we can legitimately ask what came before, or what will come after.

If the universe has a finite past, can it have an infinite future? Can the universe be semi-infinite in time? To me, this is infeasible: it would mean that we are infinitesimally close to the start of the universe, which really means that we are at the start and have made no progress in time. By the rules of probability the start of the universe should be infinitely far in our past.

Whilst the Greeks and others were concerned with origins and had their creation myths, less thought was given to eternity or the end of days. Judaism and the Talmud appear to support the immortality of the soul (there’s an interesting thread of thinking through Maimonides,

Moses Mendelssohn, and Immanuel Kant, toward what can only be described as “political correctness”). I can’t imagine an infinite life: time and change would have no meaning, fulfilment would have no meaning. We describe God as being outside of time, and so perhaps the same for our immortal souls, though it’s not clear what this might mean.

Back to the book itself. My Amazon review gives it 3 stars out of 5. There are too many ideas which don’t hang together or are irrelevant (there’s a long section on memes), too many concepts are poorly defined or not defined at all (what is “supernatural, what is a “better explanation”?), and there is too much that I simply disagree with (humans think whereas at best a machine can only pretend to think: but what is the difference?). After reading the book I realised I was probably both a “finitist” and an “instrumentalist”. Deutsch equates one with the other, and is disparaging of both.

But I’m glad I ploughed through to the end, if only because it forced me to clarify my own thinking.

Finally! This is not a rigorous scholarly work. It’s not been through the fire of review and feedback and has more holes than the proverbial Swiss cheese. Perhaps the best I can hope for is that it will prompt some discussion, feedback, and criticism.

I’d like to thank members of our community who have shared their ideas.

Referenced material 1. https://math.dartmouth.edu/~matc/Readers/HowManyAngels/InfinityMi nd/IM.html 2. https://www.aish.com/sp/k/Kabbala_2_Perceiving_the_Infinite.html 3. https://en.wikipedia.org/wiki/Russell%27s_paradox 4. https://www.myjewishlearning.com/article/immortality-belief-in-abodiless-existence/ 5. https://en.wikipedia.org/wiki/First-order_logic 6. The mystery of the Aleph: mathematics, the Kabbalah, and the search for infinity, Amir Aczel