"This confusion between qualities of animals and qualities of human society is an example of the problem of homology and analogy. By homologous traits, biologists mean those properties of organisms that are shared by different species because they have a common biological origin and some common biological genetic ancestry, and they derive from common fetures of anatomy and development. Even though they look very different and are used for very different purposes, the bones of a human arm and of a bat's wing are homologous because they are anatomically derived from the same structures and influenced by the same genes. On the other hand, a bat's wing and an insect's wing are only anagous. That is, they look superficially alike and they seem to serve the same function, but they have no origin in common at the genetic or morphological level. But analogy is in the eye of the observer. How do we decide that slavery in ants and ant queens are like human slavery and like human royal families? How do we decide that the coyness we see in people is the same as the behaviour in animals called coyness? What happens is that human categories are laid on animals by analogy, partly as a matter of convenience of language, and then these traits are 'discovered' in animals and laid back on humans as if they had a common origin."
Lewontin, R.C., The Doctrine of DNA: Biology as Ideology, 1991
The above quote from R.C.Lewontin touches on one of the most problemmatic problems of philosophy for human beings. He is using these terms in very precise ways - the biological one - where something is 'the same as' something else under certain circumstances only. Therefore the fact that we have four limbs - the same as other vertebrates - is no coincidence. In the biological sense, they are 'the same'.
When one considers less definite areas, though, things become much more problematic, and one enters areas where one's footing is less sure. The main question is - when are two things the same? Clearly, in one strict sense, nothing is literally the same as something else - otherwise there would only be one thing! However, it is equally practical to say that there are two apples lying on the table, not that there is one apple and there is another. If we wanted to distinguish the apples from a banana lying on the same table, we would feel quite justified in saying that the apples are the same, and the apples are different from the banana, even if they are both sorts of fruit. If we wanted to distinguish all three items from, for instance, the table on which they were lying, we would have no problem with saying that the two apples and banana were fruit and the table was... not!
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Take a concept such as mathematics, which deals with strictly 'unreal' things - numbers. Is the concept of the number '2' homologous, analogous, or something else? What about a sum? Or a theory concerning prime numbers? Or indeed, the whole of number theory - 'a world in itself'?
...abstract mathematical entities we think we are familiar with can nevertheless surprise or disappoint us. They can pop up unexpectedly in new guises, or diguises. They can be inexplicable, and then later conform to a new explanation. So they are complex and autonomous, and therefore by Dr Johnson's criterion we must conclude that they are real. Since we cannot understand them either as being part of ourselves or as being part of something else that we already understand, but we can understand them as independent entities, we must conclude that they are real, independent entities.
Nevertheless, abstract entities are intanglible. They do not kick back physicaly in the sense that a stone does, so experiment and observation cannot play quite the same role in mathematics as they do in science. In mathematics, proof plays that role. Dr Johnson's stone kicked back by making his foot rebound. Prime numbers kick back when we prove something unexpected about them - especially if we can go on to explain it too. In the traditiaonl view, the crucial difference between proof and experiment is that a proof makes no reference to the physical world. We can perform a proof in the privacy of our own minds, or we can perform a proof trapped inside a virtual-reality generator rendering the wrong physics. Provided only that we follow the rules of mathematical inference, we should come up with the same answer as everyone else. And again, the prevailing view is that, apart from the possibility of blunders, when we have proved something we know with absolute certainty that it is true.
Deutsch, 1997, p224
Plato...effectively denied that the physical world is real at all. He regarded our apparent experiences of it as worthless or misleading, and argued that the physical objects and phenomena we perceive are merely 'shadows' or imperfect imitations of their ideal essences ('Forms', or 'ideas') which exist in a separate realm that is the true reality. In that realm there exist, among other things, the Forms of pure numbers such as 1,2,3,..., and the Forms of mathematical operations such as addition and multiplication. We may perceive some shadow of these Forms, as when we place an apple on the table, and then another apple, and then see that there are two apples. But the apples exhibit 'one-ness' and 'two-ness' (and, for that matter, 'apple-ness') only imperfectly. They are not perfectly identical, so there are never really two of anything on the table. It might be objected that the number two could also be represented by there being two different things on the table. But such a representation is still imperfect because we must then admit that there are cells that have fallen from the apples, and dust, and air, on the table as well.
Ibid, p226
If it is indeed false, as intuitionists maintain, that there exist infinitely many natural numbers, then we can infer that there must be only finitely many of them. How many? And then, however many there are, why can we not form an intuition of the next natural number above that one? Intuitionists would explain this problem away by pointing out that the argument I have just given assumes the validity of ordinary logic. In particular, it involves inferring, from the fact that there are not infinitely many natural numbers, that there must be some particular finite number of them. The relevant rule of inference is called the law of the excluded middle. It says that, for any proposition X (such as 'there are infinitely many natural numbers'), there is no third possibility between X being true and its negation ('there are finitely many natural number') being true. Intuitionists coolly deny the law of the excluded middle.
Are drawings homogous, analogous, or what? What about algebraic expresssions which would create geeometrical patterns? How should they be read? Are they not graphical symbols too, at one level?
The reliability of the knowledge of a perfect circle that one can gain from a diagram of a circle depends entirely on the accuracy of the hypothesis that the two resemble each other in the relevant ways. Such a hypothesis, referring to a physical object (the diagram), amounts to a physical theory and can never be known with certainty. But that does not, as Plato would have it, preclude the possibility of learning about perfect circles from experience; it just precludes the possibility of certainty. That should not worry anyone who is not looking for certainty but for explanations.
Euclidean geometry can be abstractly formulated entirely without diagrams. But the way in which numerals, letters and mathematical symbols are used in a symbolic proof can generate no more certainty than a diagram can, and for the same reason. The symbols too are physical objects - patterns of ink on paper, say - which denote abstract objects. And again, we are relying entirely upon the laws of physics for sure, we cannot know for sure that the machine is genuinely rendering Euclidean geometry...
Ibid, p243
Virtual Reality - homology or analogy?
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