ENSAYO DE FATIGA POR TORSIÓN

With this in mind, let us examine Euclid’s argument once again. I suggested in Chapter 3 that a given type of representation might contribute to justification in two different ways: first, by presenting a claim in an argument; and secondly, by

mediating inference between one claim and another. We can see both these functions being discharged in the following inference:

(9) ZACD = ZABC + ZBAC

(10) ZACD + ZACB = ZABC + ZBAC + ZACB [9: by CN2, adding ZACB to both sides]

Here the sentences clearly have a presentational function. But the sentence in line 9 also mediates inference: that is, the reasoner can, just by manipulating it, reach the sentence in line 10. How so? The sentence in line 9 gives an equation of the form X = Y. Common Notion 2 (CN2) states “Equals added to equals are equal.” So adding Z to either side of the equation yields an equation o f the form: X + Z = Y + Z. Substituting back the relevant values for the variables yields line 10.

So far, then, the Leibnizian is on strong ground. This sentential inference does not require the diagram, either to present a claim or to mediate inference. So the Leibnizian has met the “sentential presentation” test for this inference. But can he meet the “no counterpart” test: i.e., can he show that this is a genuine way to follow Euclid’s argument in the way described earlier? Fairly clearly, he can. Recall that the relevant part o f Euclid’s argument is as follows:

(IV) Let the angle ACB be added to each [sc. angles ACD; and angles BAC, ABC]; therefore the angles ACD, ACB are equal to the three angles ABC, ACB, BAC.

I presented a possible alternative visual route to this conclusion in Section 2.7. But it should be evident, given (IV) above, that the sentential inference is available—and perhaps even preferable— as a reconstruction of the thinking here.

So, as regards this inference, the Leibnizian has met both tests; it is not plausible that the diagram contributes to the justification afforded by the thinking involved in at least one possible way o f following Euclid’s argument at this point. Moreover, though every inference must be examined on its own merits, this overall line of attack

substitutional inferences in Euclid’s argument, as I reconstructed it in Chapter 2, such as those to lines 3, 6, 8, 9 ,1 3 ,1 5 and 16. If he could show that these inferences also met both the tests set out above, this would go a long way to establish his overall claim that the diagram makes no justificatory contribution to Euclid’s argument.

6.6 Euclid’s Argument Revisited: 2

But now take line 7, which is reached from the diagram:

A E

C D

B

(7) ZACD = ZECD + ZACE

I reconstructed this inference in Chapter 2 as follows:

(7a) CE divides ZACD into two parts, ZECD and ZACE, without remainder [from the diagram]

(7b) The whole o f an angle is equal in size to the sum o f the sizes o f any parts into which it is divided without remainder [background assumption]

(7c) ZACD = ZECD + ZACE [7a, 7b: by substitution]

Thus reconstructed, this is a logically valid inference: lines 7a and 7b together entail line 7c (= line 7).

Again, this is a sentential presentation of the kind required if the Leibnizian is to show that the diagram is epistemically irrelevant to the reasoning required to follow

Euclid’s argument. But again, we need to recall the second “no counterpart” test: is this reconstruction faithful to the thinking required to follow Euclid’s argument? Fairly clearly, it is not. Note first that the reasoner does not seem to entertain the general thought in line 7b above, to the effect that the whole o f an angle is equal in size to the sum of the sizes o f any parts into which it is divided without remainder; and secondly, that the reasoner does not seem to do any substitutional reasoning to reach line 7—reasoning that would be required in using the general claim in 7b.

We can briefly sum up the position as follows. The Leibnizian cannot meet the “no counterpart” test in relation to line 7 by this means; a very good candidate for the required sentential inference fails the second test. Moreover, it is very difficult to see how any purely sentential inference could fare better, since doubtless the diagram would still be required to present—and warrant—line 7a, or a similar premiss. It is, then, highly implausible that the route to justified belief here is via a sentential inference; rather, it seems to be via a piece of specifically geometrical thinking that uses the diagram. This part of the thinking involved in following Euclid’s argument both uses a diagram and is sufficient to justify the reasoner’s belief state. So, if we recall the discussion o f “contributing to justification” in Chapter 3, we can say that the diagram does contribute to the justification of that belief state here.

6.7 Euclid’s Argument Revisited: 3

Is this an isolated result? No, for two reasons. First, there are other inferences in Euclid’s argument that are similar to the inference to line 7 described above. The inference to line 11 is markedly similar, for example. This was reconstructed in Chapter 2 as follows:

(11a) AC divides the angle on line BCD into two parts, ZACD and ZACB, without remainder [from the diagram]

(11b) The whole of an angle is equal in size to the sum of the sizes o f any parts into which it is divided without remainder [= 7b; background assumption]

(11c) ZACD + ZACB is the sum of the sizes of all the angles on BCD [11a, 11b: by substitution]

Again, the Leibnizian can point to a valid sentential presentation. But again, it is not plausible that this meets the “no counterpart” test. And again, there is a geometrical, diagrammatic reconstruction that does so.

Secondly, I noted above that the logical inference to line 10 met both tests; it was both sentential and, plausibly, a genuine way to follow Euclid’s argument. But we should not assume that other apparently similar sentential inferences are in fact genuine ways to follow the argument. Thus, take the inference to line 9:

(6) ZABC = ZECD [4, 5: by substitution]

(8) ZACD = ZECD + ZBAC [3,7: by substitution]

(9) ZACD = ZABC + ZBAC [6, 8: by substitution]

Again, this reconstruction takes the form of an inference by substitution on sentences. (In Chapter 2 I described an alternative visual reconstruction o f this thinking, in which the reasoner visually translates a copy o f ZECD along line BCD until it is mapped on to ZABC.) Again, for familiar reasons, understanding this as a sentential inference does not seem to fit the phenomenology. But then it is again plausible that this is just a counterpart logical reconstruction of the relevant thinking here, and the true

explanation is one that understands it as reasoning with the diagram, using a geometrical principle o f translation of angles already known to be equal.