It is an observable, testable, falsifiable fact that no computer will ever even be able to count to ten.
by Matthew Connally
Can an abacus do arithmetic? Of course not. It is just a tool that we use to keep track of numbers. Instead of using our fingers we use wooden beads on a stick: 1…2…3… But those beads don’t know what numbers are any more than our fingers do.
What if we hooked up an electric motor to a set of gears that could move the abacus beads? And what if we added a series of buttons labeled one through ten, such that pushing one of the buttons caused the machine to move the designated number of beads? Now will our little electrified abacus have done the counting? No, we would still be doing the counting, using the beads to keep track of the data.
Well what if, instead of pushing buttons we hooked up a camera lens, and instead pushing beads we hooked up a speaker? And what if we put wheels on the thing so that it could roll into another room, record the data, and declare in English, “There are ten apples on the table.” Now would our robot have done the counting?
Only if you insist that the English morpheme “ten” is in fact the number ten (in which case you would have to be reading this out loud)…or perhaps that numbers have all the physical properties of crispy, red fruits. Otherwise, to be coherent, we have to conclude that a robot is still just a tool that we use to process information—in principle, exactly the same as the abacus. Just like we can use levers to lift things we might not otherwise lift, and use microscopes to see things we might not otherwise see, and use books to communicate things we might not otherwise communicate, so also we can use computers and robots to count things we might not otherwise count—whether apples on a table or rocks on Mars.
I am going to argue that our brains are likewise merely tools, and that we who use them to count are just as nonphysical as the numbers themselves are. The mystery is not that we do not know how the brain can do things like counting to ten; rather, the mystery is that we know for absolute certainty that our brains could never do such things. That is to say that, whatever it is that does math and language, it is not that three pound organ inside your skull. Your brain cannot count any more than a radio can sing or a telescope can see or an abacus can add.
Why Can’t Computers Count?
To start with, all I am saying is that computers do not have the slightest clue what numbers are. But if all that above sounds like obscure reasoning, perhaps you’re distracted by appearances. Stop and consider the billions of man-hours required to manufacture and operate a robot. We’ve got to mine the ore for the electrical components, build factories, and invest generations of competitive research into the technology. In short, it takes a massive amount both of human labor and of human thinking to make it happen. And regardless of how the finished product may appear, to be coherent we must say that we humans are also the ones doing the math. Robots cannot count any more than the Great Pyramids can build themselves, any more than iPods can sing, any more than communication satellites can talk to us on the telephone (though the message might be relayed between them and us), any more than telescopes can see, any more than Smeagol can connive (though he sure looks real), any more than books can read, any more than pool balls and rocket ships can comprehend that energy equals mass times the speed of light squared.
And for the same reason that a computer cannot count, neither can your living, active brain. This is not a matter of semantics; this is observable, testable, falsifiable fact.
We know it by two simple observations: (1) we can observe the complete absence of any physical qualities in numbers; and (2) we can observe that numbers exist outside of our heads in the real world—that they are just as objective and useful and “real” (as real as the meaning of that word is) as a chunk of granite. Although they have no physical qualities, their objectivity is as deducible as it is for physical things.
Given these two proofs, this is a given: brain cannot perceive something (regardless of whether its objective or subjective) that cannot be directly or indirectly seen, heard, touched, tasted or smelled. So let us start with the first observation.
Numbers Have No Physical Qualities
Although we do not know what numbers are (Platonic forms? Items in a superseded ontology? Memes? Qualia?), we do know what they are not: they are not physical. This is provable: to the extent that we “know” anything, we can know this to be true.
We know this because we can translate numbers through a variety of completely different physical media—as beads on an abacus, as black symbols on paper, as sound waves, as electromagnetic waves, etc. Although none of these media have any physical qualities in common, it is possible for them to have the exact same digital information in common. Therefore, that digital information cannot be directly or indirectly seen, heard, felt, tasted, or smelled.
For example, consider the recent cinematic production of The Hobbit. It can be digitized and shot across the planet as electromagnetic waves. Then it can be translated onto a piece of plastic, a DVD. (Keep in mind the billions, even trillions of man hours required to accomplish these tasks.) Then it can be translated into light and sound waves as you watch it on a screen. Hypothetically, aliens could steal one of the DVD’s, take it to their own planet and eventually perhaps translate it. Now none of these media have any physical qualities in common. Therefore, whatever they do have in common—rational, creative communication— is immaterial.
Thus we can observe the complete absence of physical qualities in digital information. We can test any such observations by translating any information into a variety of media. And we could falsify this claim if we were able to identify any physical qualities in any particular number or piece of data.
So if we can observe the absence of tangible traits, what then are we observing the presence of? Again, we don’t know; nevertheless, it is abundantly objective.
Numbers Are Objective
Before reading on, give this riddle a shot: If apples fall in a forest and no one is there to count them, does any particular number of apples fall?
First let’s consider the no answer: until a person walks up and counts how many apples are below a tree, there is no number of apples below the tree. There are not five, six, or even zero apples there.
This answer is actually incoherent. To see this, consider what would happen if two people counted the apples but each came up with a different number? What if one person counted 17 apples and another person 18? At first we might want to say that at least one of them had made a mistake; however, what exactly would they be mistaken about? If the number depends not on objective fact but on the subjective person who counts, then there would be no such thing as a correct answer to argue about.
And all of science would be rendered null and void. I could declare that from here on out there are always exactly 5.72 apples below every single tree (not just apple trees) on the planet, and that answer would be just as acceptable as any other. In other words, nature would be incoherent. (Of course if it were truly incoherent we wouldn’t know it…)
So now let’s consider the yes answer: at any one time, whether a person counts how many apples are below a tree or not, there are a certain number of apples on the ground below that tree. The number is just there, waiting to be perceived, translated—for example, into English (“seventeen”) or German (“siebzehn”), etc.—and used.
Now we could indulge all sorts of rhetorical gymnastics about epistemology, ontology, and other relentlessly esoteric ideas. Nevertheless, in the end we would still be left with one of only two choices: numbers are either subjective or objective. Any attempt to find a third option is just a frolic in the fog of grey (which is defined by the dichotomy of black and white). And regarding these two choices, all science operates on the assumption of the later: numbers are objective. To the extent that we “know” anything, to that same extent we know this is true.
Mathematicians don’t author proofs; instead, they discover them. Astronomers don’t author how many stars and galaxies are out there; instead, they seek to discover what’s out there. Physicists don’t author the force of gravity; instead, they measure it. All these rational, creative observations—the deep and complex equations, the astonishingly complex order, the breathtaking explanations—are 100 percent objective.
Philosophers can pretend to doubt this assumption but, in truth, they cannot actually doubt it. For it provides the very context by which the word “doubt” has any meaning—just like the word “meaningless” is only coherent in broader context of meaning. No amount of rhetorical gymnastics can defy the fact that all scientists and engineers assume these things to be true, and operate with them in faith.
Furthermore, to magnify the power of these observations, we can deduce that all such meaning (i.e. objective mathematical meaning) always precedes its medium. For example, long before we built the Mars rover Curiosity, we wrote up a plan for it. That plan was edited and edited again, many times, before it was ever translated into metal parts. Similarly, before an apple grows there is a DNA code for it. The same is true for all rational, creative communication: first comes the data, followed by its medium. The data can exist without the medium but the reverse is not true. The concept of infinity, for example, and our use of it in calculus—without which we would have zero modern technology—cannot ever have any concrete medium at all. It is, ipso facto, immaterial.
All that is to say that numbers are objectively out there, everywhere, in every quanta of the cosmos. Although we know that there are matters of uncertainty and randomness, these are only coherent in the broader context of what is certain. I can be absolutely certain that you are a rational, trustworthy person, yet still have absolutely no idea what you are going to say next. Your words might be (literally) creative!
So like it or not, perceive it or not, translate it or not, the very hairs of your head are all numbered. Indeed, every quanta of the cosmos is a medium for rational, creative information.
Who Can Count?
Again, it is simply incoherent to argue that a robot can count—as incoherent as arguing that the number ten is red.
But what about the human brain? Must we not also conclude that it cannot count? In all of his writing about the mind, the brain, and language, Harvard Professor Steven Pinker addresses our amazing ability to count only once:
“Humans, like many animals, appear to have an innate sense of number, which can be explained by the advantages of reasoning about numerosity during our evolutionary history. (For example, if three bears go into a cave and two come out, is it safe to enter?) But the mere fact that a number faculty evolved does not mean that numbers are hallucinations. According to the Platonist conception of number favored by many mathematicians and philosophers, entities such as numbers and shapes have an existence independent of minds. The number three is not invented out of whole cloth; it has real properties that can be discovered and explored. No rational creature equipped with circuitry to understand the concept “two” and the concept of addition could discover that two plus one equals anything other than three.” (The Blank Slate: The Modern Denial of Human Nature. Steven Pinker, pp. 192, Viking Penguin, 2002.)
Pinker certainly is correct in observing that counting appears to be a very easy, simple task. Nevertheless, it still makes no sense to say that circuitry can perceive numeric entities (or Platonic concepts or whatever), unless those “entities” have some physical qualities. Is that what Pinker was implying when he talked about the number three having “real properties”? That would be exactly like declaring that the number ten is red!!!
Instead, we must say that we use our circuitry to perceive the entities. We use our brains to process data taken in through the five senses—in principle the same we use abacuses and computers to process data. They are tools.
So if our brains are tools, then what are we? “I count, therefore I am…what?!”
We don’t know. But we do have very compelling evidence here for what we are not: we are not physical. We are as immaterial as numbers are. Consider that in the discussion above about The Hobbit, there is something else all those media have in common: authorship. They do not have the brain in common. (That is to say, for example, that there is no organic gray brain matter on a DVD.) But they do have human authors in common. What is an author? It is something about as nonphysical as is rational, creative communication.
And the mind-over-matter conundrum remains as mysterious as ever. Neither Pinker nor any other Naturalist can pick up this gauntlet. Indeed, Albert Einstein called our ability to perceive data “the eternal mystery of the universe.”
Let us stay transfixed on this mystery, and not be distracted by all the amazing technology—the robots, the artificial intelligence (read: not real, illusionary intelligence), CGI, etc. Sometimes you can listen to brilliant voices singing to beautiful music, yet the actual words of their songs are pure nonsense. Let us ignore all this majestically beautiful babble, and instead seek to hear from the Author of life.
Matthew Connally is not a scientist, but this debate has nothing to do with the facts of science. Rather, it is about rhetoric and the meaning of words. He can be reached at firstname.lastname@example.org.