I’m still contemplating a few issues raised or brought back to memory by my recent reading of The Meaning of Truth by William James. The ideas of William James put pressure upon my own ideas, a form of updated Thomism, just because he knew how to respect empirical reality without being a reductionist but also, mistakenly in my opinion, without being willing to posit a world of which his empirically observed things and relationships are a part. So it is that I’ve returned to vague thoughts about that precise subject of mathematics. Can we make up a large number simply by writing down some symbolic representation and inventing a name for it? Is it meaningful?
The name googol started as a whimsy of sorts: it’s defined as (10^100) or a 1 followed by one hundred zeroes. It was part of an educational effort, the discussion of the difference between very large and infinite. In that context, the meaningless number acquired a good enough meaning to justify its use. More reasonably, large numbers might fall out of proofs involving questions such as: at what point do prime numbers become as common as they theoretically should be? (In the range of small and readily accessible numbers, prime numbers are too sparse according to number theory.) Large numbers can also fall out of calculations such as time and quantum state calculations involving a number of particles similar to the number believed to exist in our universe.
Some sort of objectivity in naming a large number or writing it out in symbols seems to be required if only for intelligent discussion with other human beings. Otherwise, pure whimsy in the realm of numbers and other matters of cognitive substance could turn certain discussions into playground arguments: “Yeah, well I’ll take your number and multiply it by a million and my number’s a lot bigger.”
I think there has to be some sort of operational meaning to a large number, such as the above example where very large numbers sometimes pop out naturally in various sorts of proofs or explorations in number theory or explorations in theoretical physics. Very large numbers can also arise in studies of computer science and cryptography. An interesting example of large numbers arising in the theory of recursive functions, very important in mathematics and computer science, is the Ackermann function which rapidly produces numbers which are very large by a remarkably simple calculation. The power of this function to generate large numbers is somewhat surprising and has been explored a bit in various modified forms. A reader with even a casual interest can read a little more in the way of background in this article on large numbers.
I return to the more general issue of truth having an experienceable nature as William James claimed in The Meaning of Truth and I discussed in a recent essay, What Can Be Experienced?. This is the issue: if truth is experienceable, if it must have some sort of operational meaning for human beings to grasp it, does sanity require boundaries on our imaginations? I’m most certainly not reducing truth to an operational form, but I am saying that human beings need something to grab hold of for any sort of truth to be such in our sorts of minds. Our minds are relationships and need to have a true relationship to even a number for it to reside in these minds. Put in other terms, we can understand the functional relationships which recursively generate very large numbers even though the sizes of those numbers are far beyond the size of our minds. In a similar way, we can understand entities too complex to hold in our minds, such as a non-symmetric black-hole or a human community or a strand of DNA in a plague bacterium. We can also understand things which don’t happen to exist but maybe could exist in another plausible universe or in our universe if it had developed differently.
We can even picture a Jabberwock, at least with the help of the artist Sir John Tenniel. Though that poem was no more, and no less, than a wonderful piece of whimsy, the strange creatures and even the meaningless words have some meaning to us. They can be experienced in a meaningful way, but maybe words which `fit’ but have no meaning are little different from a large number created by simply piling up exponents? In response, we could point out that many long extinct creatures, and a few discovered deep in the ocean, are as strange and as nasty as the Jabberwock as imagined by Lewis Carroll and his many readers.
That nonsense poem about the non-existent beast began as a different sort of nonsense poem, a parody of sorts:
Stanza of Anglo-Saxon Poetry
Twas bryllyg, and ye slythy toves
Did gyre and gymble in ye wabe:
All mimsy were ye borogoves;
And ye mome raths outgrabe.
It was apparently written to poke a little fun at the efforts of modern English-speakers to read older versions of their own language. It is historically-grounded whimsy, not an effort to shape something allegedly new by using some selected facts of human history to invent some nightmare world, such as a bar in which we can watch the interaction of races which evolved on different star systems and yet can easily communicate, heck! — they can eat the same foods and drink the same intoxicating liquors.
So, can we simply string together exponents: 10^(10^(10^(…))) and call it a number? Can we simply make up other universes or entire infinities of other universes?
There seem to be two requirements to establish the legitimacy of efforts of whimsy, fantasizing or invention if you will. The first is that, like the `Anglo-Saxon Poetry’ of Lewis Carroll, it has to be plausible in some way, even if just by archaic appearance or sound. The second is that it has to be experienceable in some meaningful way, not a way that is fundamentally dishonest. Incompetence in both content and execution should be eliminated, will be eliminated by time alone. Schlock of the sort dominant in science-fiction — there are a few worthwhile works in those piles of strange and smelly things — will prove to be ephemeral in its appeal since it stands in some dreamworld of one generation and not in some objectively accessible understanding of historical and scientific reality.
I don’t think we can simply string together exponents: 10^(10^(10^(…))) and call it a number. Some sort of operational meaning is required.
I don’t think we can simply make up other universes or entire infinities of other universes. Some sort of grounding in historical reality is required.
There are other streams of events which might have occurred in our universe and there are other possible universes, but we have to imaginatively work from the one universe we can directly perceive and explore. Our thoughts about universes and their various narrative streams: astrophysical or biological or historical, should be plausible in light of what we can perceive and explore.
The various confusions about proper use of the human imagination do little harm in mathematics and other fields of science as currently defined because of the forms of disciplined peer-review which tend to dominate, if imperfectly so. This could change. Even mathematics could lose its fortifications against irrationality, as we modern men settle down in that state of barbarian childhood seen and foreseen nearly a century ago by Jose Ortega Y Gasset in The Revolt of the Masses which dealt with the failure of the masses of peasants and their urban equivalent to grow into the greater realms of civilization opened to them in the modern era. Ortega was actually most upset by the failure of the cultural and social elites to teach those liberated peoples, to help them to integrate themselves into the civilization of Moses and Plato and St. Paul and Dante and so forth rather than simply becoming part of a mass of consumers and seekers of forms of entertainment appealing to our raw animal desires. In fact, those elites have themselves sunk to that low level of cultural and spiritual vulgarity.
I’ll make one final point about an inconsistency in the thoughts of many fans of so-called science-fiction and fantasy. A book or movie can be criticized for taking too much imaginative liberty with the facts and strongly established theories of physics and chemistry and most other fields, though a great deal of liberty is allowed in the realm of biology. Almost complete liberty is allowed to differ with the human nature we know mostly through history even in this age of neurobiology. History teaches some harsh lessons about the political and social and moral aspects of human nature, the only rational nature we know about. Science-fiction often seems to imagine, in what I’d call a diseased manner, that the developmental and evolutionary processes of this universe could be projected ahead so that we can imagine the sort of creature which could be peaceful in a way which is in contradiction to the nature of such a universe.
We could speak in Christian or Darwinian terms, or even mixed terms as I often do, but it’s hard to sanely imagine a world shaped by evolution being peaceful or orderly in the way we often desire and some think possible to realize just because they desire it and think they can imagine that peaceful world so that it can be realized despite the nasty lessons of human history. This is a mistake being made by not just science-fiction fans but by many Christians including prominent leaders.
I’m currently reading some histories of the ancient world and one lesson of those histories is the tendency of multicultural civilizations to break down into wars of one group against each other even for relatively trivial reasons. The Greeks were a homogeneous people and the Greek colonists in Italy who were descended from Corinthians couldn’t live in peace with Greek colonists in Italy from the island of Samos, but neither could the parent-polities in Greece live in peace. This doesn’t mean it’s impossible to form a multicultural society which is peaceful over the long-term but to merely preach that peoples from a variety of alien races can automatically get along so long as they go along with the (undescribed) program is morally irresponsible and intellectually empty. A desirable goal, that of a human race where each loves the other and all groups can co-exist next to each other, isn’t an intelligent plan of action, especially when that goal is probably unreachable in this mortal realm. We should remember the teaching of St. Thomas Aquinas about the intentionality which forms human moral nature and, hence, human communities. Intention isn’t some subjective desire or motivation or goal, but rather a growth process, more often than not a slow and sometimes painful growth process.
We should be careful and morally responsible in our use of our imaginations and never assume that our imaginations correspond to reality. In Human Moral Nature: An Overview, I discuss my understanding of the proper way to form abstractions and proper use of imagination is an integral part of this way. Part of that essay presents a sketch of an imaginative journey from the concrete reality of an apish physical creature up into more abstract regions of being where moral paths and Einsteinian spacetime paths are the same sort of entity. This is one way to define proper use of the imagination, you can stray from what has been concretely realized but you must stay in the domain of what is possible when considering the abstract stuff from which that concreteness has been shaped. We must also consider the evolutionary and developmental processes by which that abstract stuff is shaped into concrete things and relationships, including creatures with community lives.
To act as if something is possible just because you can string together words describing some sort of alleged good, can be morally irresponsible. Even to act as if the possible can be directly realized — and by late this afternoon — is just as morally irresponsible.