Limitaciones de un control PID

There is an obvious rejoinder to this line of thought. This is that the view of Ross’s Plato ought to hold that the generalisation here is by induction on several diagrams. Maybe it is like this: the reasoner uses the rubric to draw a particular diagram of a triangle. She follows the Demonstration and observes that, as far as she can tell, the diagram is triangular, and the internal angles of (the relevant portion of) the diagram sum to two right angles. She then constructs another such diagram, and observes that it too seems to have the same property. On this basis, she formulates a general hypothesis: that every triangle has the angle sum property. This hypothesis is then tested by her and others by a process o f experimentation, in which she draws and examines further diagrams, or imagines certain similar figures. (Maybe she re-runs the argument for diagrams of equilateral, isosceles and scalene triangles.) The inference to the general conclusion is, then, via an inductive generalisation from sense-experience.

This is more plausible: let’s call it the Inductive View for convenience. The

Inductive View seems to create a tight connection between the answers offered to the two questions above. Why should one think the diagram contributes to justification? On this view: because the diagram presents evidence for the reasoner’s belief. Why think the justification is empirical? On this view: because it relates to the diagram, a physical object, and the visual information that the reasoner derives from the diagram. Indeed, as noted in the previous chapter, it might even seem as though the Inductive View is obligatory for someone who thinks that the diagram makes a justificatory contribution.

But here is a difficulty: the reasoning process just described goes well beyond the kinds o f reasoning discussed in Chapter 2. There it was noted that a reasoner does not need to consider more than one diagram in order to derive apparent justification for her belief; where she does do so, moreover, it seems to be to confirm that she has implemented the construction process correctly, not to justify her belief in the

conclusion o f the argument. Moreover, in the kinds o f reasoning described in Chapter 2 there is little that can be plausibly described as evidence-gathering. For example, it

or angles that seem or are perfectly s h a r p . A n d the reasoner does not need to scrutinise the diagram carefully at the outset, or to monitor it during the course of the reasoning, to ensure that it or its properties have not changed. These considerations undermine the claim o f the view of Ross’s Plato to be an account o f the type(s) of reasoning that we have taken as our target.

Such considerations are hardly conclusive: perhaps we were just mistaken about the phenomenology o f the reasoning noted earlier, or the physical state o f the diagram is so obvious as not to need checking. But they can be strengthened by considering some o f the properties of the beliefs acquired as a result o f following Euclid’s argument. I noted in Chapter 2 that, among other things, the reasoning involved in following Euclid’s argument appeared to give rise to feelings o f accessibility and certainty in a reasoner. It seemed as though a reasoner who follows the argument can acquire a belief in the conclusion quickly; that is, that the transition from

understanding the conclusion to believing it is a short one (possibly, for some

inferences, even a phenomenologically immediate one). And it seemed as though that belief amounted to a strong conviction, a feeling that matters could not, or not easily, be otherwise.

Can the Inductive View explain these feelings of accessibility and certainty? Take the question o f accessibility first. In many inductions to new belief, the reasoner does not make a rapid transition from understanding a claim to belief in it. Rather, after

following an argument through for a few early cases, she entertains a general claim as a hypothesis; she then comes to form a (frequently tentative) belief in the hypothesis after further experimentation. There are at least two types o f case, however, in which the transition to belief might be very rapid. In the first, a belief is formed via an inductive generalisation and becomes entrenched and familiar. New data then

emerge, where it is obvious that these fall under the existing hypothesis. The scope of the existing belief is slightly extended to cover the new data, and this process can be very rapid. A second type of case is one in which an entirely new belief is formed, but it is based on an overwhelming preponderance o f evidence on one side, and little or no recalcitrant evidence on the other. Is our target reasoning one o f either of these

I use “sharp” to describe the appearance o f an angle where the component sides appear to overlap at exactly one point; an obtuse angle can be sharp in this sense.

types o f case? Surely not. Belief in Euclid’s conclusion is not a mere extension of existing belief, and there is no preponderance of evidence: quite the contrary.

Now take the question o f certainty. Some inductive generalisations are such that, other things being equal, the strength of the belief produced is rationally positively related to the number of instances in the sample that constitutes the evidence. A belief that all sheep are woolly can thus rationally be strong for someone who has encountered a large number of woolly sheep. And some inductive generalisations are such that the degree o f belief resulting from them grows incrementally with new (positive) evidence. The degree o f belief may initially be low, especially in the early stages o f experimentation, and so very sensitive to new evidence. The subsequent rate o f growth in degree o f belief may be large for further increments of (positive) evidence, and it may even start to decline, but it is not normally negative. But if there is an inductive generalisation here, it is not like one o f these. For the evidence in this case, the sample o f beliefs reached by drawing different diagrams, is likely to be rather small. If it is not, it will not respect a fair description o f what actually takes place; the reasoner does not, as we have seen, normally draw many different diagrams or visualize many different figures. If the belief here is generated by a generalisation like those mentioned above, then given the paucity of evidence, the degree of belief generated should be relatively small. But in the case we are discussing it is not small; on the contrary, it typically amounts to a strong conviction, as noted. Moreover, it is not low at the outset. In this case, the degree of belief in Euclid’s conclusion is equal or close to 1; this degree o f belief is achieved very soon after the reasoner concludes the argument; and it shows little or no sensitivity to new “evidence”.

Again, these considerations are not conclusive: maybe there are suitable models of induction that can explain these phenomena. But we can get a better understanding of what is at stake here by exploring the sharp contrast between the kinds of reasoning we saw in Chapter 2 and a process o f mathematical reasoning that is clearly empirical. Imagine the reasoner who is unaware o f the angle sum law and is asked to measure a diagram o f a triangle with a protractor, as part of an effort to see if the internal angles o f triangles always sum to 180°. Here we might well see all the factors mentioned above in play: very precise drawing of the diagram, careful attending to lines and

error, the desire to draw many different diagrams to confirm the hypothesis,

increasing confidence in the hypothesis, and perhaps final strong, but still conditional, belief. None o f these needs to be in play in the kinds o f reasoning surveyed in

Chapter 2.