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Impossibilities Chapter 7 Impossibilities By rights, nothing in this chapter should exist. But a chapter on impossibilities is a necessity in a book on paradoxes. We begin by looking at pretend impossibilities, illusory paradoxes that are readily resolved by careful clarity. We next look at genuine impossible objects, or rather actual drawings of impossible objects. We also pay attention to some impossible intentions. Impossibility is sometimes due to language, sometimes to our conceptual powers; it is never grounded in the impossible object itself, which necessarily fails to exist. We end by considering a mathematical necessity so strange it seems it must be impossible. 1081_BBo-Paradoxes_132-147.indd 132-133 04/02/2013 15:00 VARIETIES OF IMPOSSIBILITY A QUESTION FROM WITTGENSTEIN To be impossible is almost a contradiction in terms. If something is What time is it on the sun? Is it even (merely) false that it is prior to 4:00 am on possible for this question to have an the sun, then is it not 4:00 am or later impossible, it cannot be. So how can we speak of being impossible at all? answer? Can it be morning on the sun? Is it there? Impossibility is rather a form of necessary non-being. The impossible not 4:00 am on the sun once a day? If it is does not exist because it cannot exist. The impossible is a species of nothing. And if the impossible is nothing, surely nothing is impossible. proposition conjoined to its negation. Each Decide for yourself: is consistent by itself, but they mutually exclude each other so can’t both be true. A “Virtue is triangular.” On What There Isn’t So it is possible to discern various types of married bachelor offends by being both There is an old philosophical question impossibility and, as we shall see, there are unmarried, as are all possible bachelors, and “Breakfast is the first meal to become about whether holes exist. A hole is not so innumerable species of impossible objects. yet married. Satyrs, centaurs, and sphinxes president.” much a something as a nothing—a lack, an Impossibility is necessarily diverse. One are all impossible monsters composed of absence, something missing. But you can might almost say that many forms of the possible parts. In this way composite These statements, you will no doubt agree, count holes, which means that many exist. impossible are possible, and here are some. impossibilities seem to be impossible wholes are not true. But would you call them If many holes exist, why shouldn’t one composed of possible parts. false? To call them false has seemed to exist? If there are more holes in a piece of Practically versus Could there be an inherently some philosophers to be giving them more Swiss cheese than there are in this Inherently Impossible impossible object not composed of possible credit than they deserve. These sentences argument, then that means there must A useful distinction can be made between parts? Or does it take at least two commit category errors; they are too exist at least one extra hole in that cheese what is practically impossible and what is possibilities to be put together incorrectly outlandish even to be called false. For that is not in my argument. And if that inherently impossible. What is practically in order to make an impossibility? One instance, the first sentence applies a hole exists, so does every other hole. impossible today may be routine tomorrow, must wonder whether simple impossibility concept from one domain (geometry) to a The impossible is very much like a for instance if technology develops. Travel is even possible. But if a simple concept from a totally different domain hole. Impossibilities may literally be to the moon was once impossible, but impossibility is simply impossible, that (ethics). This sentence is in fact a stock nothing, but they can be counted and set ingenuity, determination, lots of money, and would surely prove to be the case in point. example of a meaningless or nonsensical out in a bewildering array of varieties. One a competitive spirit changed that. It has, And, with that mind-bending thought, you claim (meaning it cannot be significantly is tempted to say that impossible objects however, always been logically possible. may be relieved to read that simple said to be either true or false). come in many shapes and sizes; the Our concern in this chapter is the more impossibilities are too complex to be Against this pro-nonsense position it trouble is they don’t “come” at all. difficult impossibilities that are not due to further considered here. may be argued that, if the above claims And yet the impossible can not only be practical obstacles. We are looking for the were false, they would not be false by imagined, it can be discovered. For inherently impossible—that which cannot The Meaningless versus accident. Indeed, if they were false, they example, in the field of physics there is the be achieved or realized in any way, and the Impossible would be necessarily false, and so law of conservation of energy (energy never can be. The attempt should be made to distinguish impossible. Moreover, those who try to cannot be created or destroyed) and the between the meaningless and the distinguish the meaningless from the law of special relativity (it is impossible to Simple versus Composite impossible. The meaningless (also called impossible in this way will find themselves exceed the speed of light); also, in the field Impossibility the nonsensical) is defined in some regions uncomfortably committed to the claim that of mathematics, we saw earlier that Gödel Most impossibilities are made up of of philosophy as any claim that can be it is impossible that it should be true that proved a complete theory of arithmetic is inherently possible parts that simply cannot neither true nor false. The impossible, in virtue is triangular. The distinction itself impossible (see p. 59). be put together in the way proposed. For contrast, is whatever is necessarily false. becomes meaningless, and the nonsensical example, a self-contradiction is a is seen as simply the impossibility of sense. 134 Paradoxes Impossibilities 135 1081_BBo-Paradoxes_132-147.indd 134-135 04/02/2013 15:00 TRANSGRESSIONS THROUGH definition this is the point at which we can mate with people of the previous say that distinct species have evolved. millennium, who in turn could mate TRANSITIVITY Now imagine a biologist comes along with people who lived in a millennium and takes a number of specimens from previous to that. Assuming transitivity, Imagine a businesswoman who is detained on business in Boston, and each lake. He returns to his lab and places one can appear to prove that there was the different fish in the same tank. In the never a distinct species from which misses home and family in Cleveland. Homesick, she sighs: “If I were course of his experiments he observes the human beings arose. not in Boston, I would be in Cleveland.” We may assume she is speaking following: Intuitively we know that this can’t be the case, but what has gone wrong? Is the truth. Now surely it would also be true for that person to affirm the Group A can interbreed with Group B. our definition of a species (based on following sentence: “If I were in Alaska, I would not be in Boston.” Group B can interbreed with Group C. interbreeding) at fault; is the set of Group C can interbreed with Group D. observed facts a biological impossibility; or Group D can interbreed with Group E. is the principle of transitivity worthless? This innocuous conjunction leads by an Well, transitivity is safe, but it is misapplied Group E cannot interbreed with Group A. Is the case laid out above an a priori apparently indubitable principle to an here. Transitivity assumes that the B clause argument against the evolution of species? impossible conclusion. The principle, (not in Boston) means the same thing in We can apply transitivity to this to say that Well, in a word, yes—albeit a very bad one. known as “transitivity,” takes the form of both premises; however, the vagaries of if Group A can interbreed with Group B, In reality, what it shows is that our valid argument: if the two premises are language mean it doesn’t in this case. and Group B can interbreed with Group C, chosen definition is inconsistent with true, the conclusion must also be true. Intuitively we understand that a more then Group A can interbreed with Group evolution. Despite its intuitive appeal, Check for yourself: complex analysis is required than simply C. By applying transitivity a few more times the definition is an abstraction with at forcing these sentences to fit the principle you can reach the conclusion that Group A best a local truth. Species in reality are If A, then B. of transitivity. can interbreed with Group E—but this more like individuals (like particular If B, then C. contradicts the observed fact that Group E twigs on the evolutionary tree) rather Therefore, if A, then C. A Biological Impossibility? cannot interbreed with Group A than discrete classes. A further example of transitivity going awry The same conundrum can be put in can be found in the following example. another way, considering groups separated We start with an, admittedly crude, not by physical barriers but by the passage definition of a species involving the ability of time. People of today could in principle of two individuals belonging to it to interbreed. (This is based on biological principle, so irrelevant factors like age, opportunity, and inclination are ignored.) Now imagine that a particular species of fish populates a lake.
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