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What the Tortoise did not say to Achilles

August 31, 2008


The write-up here is a dissection of the intriguing dialogue between Achilles and the Tortoise in an attempt to demystify the meaning, implied or otherwise, of what is being said by the Tortoise and what is being taken in by Achilles. To appreciate this write-up it is helpful, if not essential, to read the actual dialogue written by Carroll.  



“(A) Things that arc equal to the same are equal to each other.

  (B) The two sides of this Triangle are things that are equal to

      the same.

  (Z) The two sides of this Triangle are equal to each other.”


The tortoise is a trickster…



“(C) If A and B are true, Z must be true.”


Consider the two propositions:


“(A) Things that are equal to the same are equal to each other.

  (B) The two sides of this Triangle are things that are equal to the   



The tortoise describes his position thus:



“And might there not also be some reader who would say ‘I accept A and B as true, but I don’t accept the Hypothetical ‘?”


“And neither of these readers,” the Tortoise continued, “is as yet under any logical necessity to accept Z as true?”


Note that here the Tortoise accepts that Euclid’s treatise establishes the requisite “logical necessity” to accept C. And that since Euclid isn’t even born there is no such “logical necessity” existent. The fact, however, is that Euclid simply stated what always existed – the “logical necessity” (discussed below).


“Quite so,” Achilles assented.


This is where Achilles commits the biggest blunder.


The tortoise continues:

“Well, now, I want you to consider me as a reader of the second kind, and to force me, logically, to accept Z as true.” 


Z is, in fact, true for it contains no new information that is not included in A and B. Z essentially restates part of A and B in one proposition but the fact of the matter is that the reverse process – given Z, obtaining A and B – is not possible for lack of sufficient information in Z. That A and B are true means Z is true for there is enough information, given A and B, to arrive at Z. Z is a subset of the union of A and B, in fact an intersection of the two. Therefore there is no need to grant “If A and B are true, Z must be true” for you have, in you acceptance of A and B, accepted Z. Stating Z separately is just to bring to notice a fact that is inherent is A and B – a fact that isn’t a new reality in that it has no information that is not already in A and B. You have to grant propositions like A and B and not the validity of the fact that A and B imply Z for in granting A and B you grant Z and you only contradict yourself if you don’t grant the fact that “A and B imply Z”.


That Z holds given that A and B hold is not to say that A and B hold. It just says that if A and B hold Z also holds which is NOT a Hypothetical in that (as I said) Z presents no new information but that which is already in A and B. To put it more clearly: Z is a necessary but not sufficient condition for A and B to hold. That “A and B imply Z” describes the dynamics of how A and B, given that they hold, contain some result that is intrinsic to them but which has nevertheless been explicitly stated in Z. A thing that contains something cannot but contain it. The purely logical argument aside, to believe A and B you have to see it for yourself through experiment or observation and then establish their truth value. Logic can’t give you any new information about the world. It can only, given some information, tell you what truths are contained in the information given. These truths obtained through logic can in themselves contain some other truth. It’s like this: you begin with a set (call it O) of simple propositions (let’s say you have verified these by experiment/observation and that you, therefore, grant them). Given these, you state another set of truths (call it P, the set of primary truths) contained in these propositions but not stated explicitly. Now that you have set P, you check if there are any truths implicit in P but not stated explicitly elsewhere. Call this Q, the set of secondary truths. Check Q to obtain R, the set of tertiary truths and so on… note that this chain can go on if you don’t state explicitly in P all the truths not explicit in O. If you do state explicitly in P all the truths not explicit in O the chain ends at P itself. If not you construct Q and if in Q you state all the truths not explicit in O and P, the chain ends at Q. Otherwise, as I said, it goes on… it ends at the set in which all the truths not explicit in all the previous sets have been stated. This, in fact, means that no matter how farther down the chain you go, the only TRUTH the sequence of sets expresses is the truth of set O. You can’t state anything that is not in O using logic. To state any truth that is not a part of O you have to establish its validity not through logic but through observation, the possible reasons for it, and experiments that can justify your reasons.


The fundamental principle, therefore, is observation, reason, and experiment – the scientific method. Logic is simply a construct that, given a range of possibilities, describes how, given one or more of these possibilities, you can explore the truth inherent in them. Logic can’t substitute the scientific method for it is a subset (and not a proper one at that) of the set “the scientific method”. In fact, much of what passes for “sciences” not based on “the scientific method” is but a set of unchecked assumptions, given which, the practitioners of these “sciences” establish the logical conclusions thereof. They make observations, they reason, but they are not willing to experiment. If they do experiment even if not to check the validity of the hypothesis itself but that of the logical conclusions thereof, they pass for scientists; for much of modern science progresses by making hypotheses (this is the set O) and checking if the logical conclusions thereof (P, Q, R… ) hold. And, say, if you establish all the logical conclusions P, Q, R… through experiment you have established all possible truths that follow from O – apparently, then, since you have established everything that could be concluded from O these conclusions might be used to conclude O itself. Since you have checked these conclusions by experiment you have, in effect, checked O, your hypothesis by experiment: it should therefore be possible, in principle, to check your hypothesis directly through experiment. And if it is not then you haven’t really concluded all that could have been concluded from the hypothesis – some truth still remains that hasn’t been discovered and established experimentally.


That is the scientific method – assumptions (backed by experiments) that hold until an experiment contradicts them. And whenever a contradiction is observed we have definitely found something unknown to us – we seek to explain it, develop theories, modify assumptions, make predictions, and continue to have an increasing faith in our theory until an experiment contradicts it and we have to rethink our theory again.


P.S.: This write-up has been reworked from a discussion on ‘Rationality’ that I initiated on Vimarsh and available here:


Do you agree with the analysis? Comments on what you think the Tortoise did not say to Achilles or counter comments to my perception of the dialogue between the two are most welcome. 



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