# LAKATOS PROOFS AND REFUTATIONS PDF

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One of Lakatos' goals in writing this dialogue was to argue that mathematics is The full dialogue is available as a book called “Proofs and Refutations” (which. Cambridge Core - Philosophy of Science - Proofs and Refutations - edited by Imre Edited by Imre Lakatos, John Worrall, Elie Zahar . PDF; Export citation. Library of Congress Cataloging-in-Publication Data. Lakatos, Imre, – Proofs and refutations: the logic of mathematical discovery / Imre Lakatos.

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Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the. PDF | On Jan 1, , Richard A. Demillo and others published Review of and refutations: the logic of mathematical discovery" by Imre Lakatos. Proofs and Refutations is a book by philosopher Imre Lakatos expounding his view of the . Create a book · Download as PDF · Printable version.

Yet, another counterexample was given which contradicts the modified Lemma iii. So, Lemma iii again needs to be modifed to save the proof. Alternatively, we can modify the terminology: we call boundary triangles those whose removal does not disconnect the network.

As previously observed, one cannot remove the triangles from the inside of the lat mapping. Thus, the role of this counterexample is that of a positive heuristic as it helped us to prove what we set out to prove.

The underlying reason for this is the fact that the polyhedron was manipulated adopting a topological point of view, disregarding its metric qualities and quantities. Whatever the size of an edge or a face of the polyhedron, the proof still applies. But, what makes this specific proof applicable? Thus, the topological invariants of the network remain the same. Recall that a homeomorphism or topological isomorphism is an isomorphism which preserves topological properties.

A homeomorphism, thus, is a bijective and bicontinuous function. As a result, homeomorphic topological objects have the same topological properties. For instance, a sphere and a torus are not homeomorphic, thus do not enjoy the same topological properties. In conclusion, this lemma shows that the simple topological object is preserved under homeomorphism, and in this particular case, the homeomorphism is the function of removing a triangle from outside of the planar network.

Further, this mapping does not violate the simpleness of the topological object.

Essentially, global counterexamples, outside of any reference to the proof of the conjecture, demonstrate that it is false. However, the first global counterexample which was given in PR was also local.

Hence, it is a counterexample for the first lemma, since the nested cube cannot be stretched onto a plane after a face has been removed from the inner polyhedron. Thus, they altered some definitions of polyhedra in order to save the conjecture, and suggested the following definitions.

Definition 1 A polyhedron is a solid whose surface consists of polygonal faces. Definition 2 A polyhedron is a surface consisting of a system of polygons. First, note that in order to deal with the case of the nested cube, the protective belt of the hard core of the theory of polyhedra was employed. This opened up a discussion on the definitions of polyhedron in order to save the conjecture. The underlying motivation was the following.

In order to exclude the nested cube as a freak, or monster as Lakatos himself called it, it must be proven that the nested cube is not a polyhedron at all. To accomplish this, one can define the polyhedron as a proof generated concept. As long as the students stuck to the Cauchy proof, a rigorous and precise definition of polyhedra was needed and thus this definition was derived from the proof in such a way that the definition would not conflict with the proof and any of its lemmas.

As Lakatos clearly and radically considered these concepts, we will not repeat them here.

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However, we note that, for Lakatos, the proof was in fact needed in order to shape the notion or the definition of the polyhedra. Notice that this revision of definitions seems to be an ad hoc. After each counterexample, one might need to revise the definitions in order to exclude the monsters. As might be expected, new counterexamples emerged immediately after the new definitions were introduced.

In PR, they were labeled as Counterexamples 2a and 2b. Observe that both counterexamples fit both definitions. In order to exclude these counterexamples, an immediate revision of the definitions took place in a rather ad hoc manner. Counterexamples 2a and 2b were presented in a way as to show that those definitions were much too restrictive, i. Definition 3 A polyhedron is a system of polygons arranged in such a way that 1 exactly two polygons meet at every edge and 2 it is possible to go from the inside of any polygon to the inside of any other polygon via a route which never crosses any edge at a vertex.

The result was the Perfect Definition, which we denote by Definition P. We believe that this is the reason why PR contained further material.

The question then becomes, why did Lakatos include this ad hoc definition which was indeed very narrow, restricted and heuristically weak? In our opinion, the underlying reason for the introduction of Definition P.

Obviously, in the very narrow domain described by Definition P. Moreover, it should also be underlined that this definition was proof-generated, and thus came after the proof. At the first glance, Definition P.

However, when considering the fact that the Definition P. We believe that this move was a very significant accomplishment of Lakatos: he turned a deductive-looking concept into a proof generated concept by revising the heuristic presentation. It was proposed to refute Definition 3. So, by the rules of negative heuristics, Definition 3 was revised to exclude the urchin.

Definition 4 A polygon is a system of edges arranged in such a way that 1 exactly two edges meet at every vertex, and 2 the edges have no points in common except the vertices. This definition was again a product of negative heuristics, since it excluded a clause of the previous definition and restricted the concept of a polygon even further.

The lines which intersect in two dimensions may not intersect in three dimensions. One can approach this naive observation from low-dimensional topology or even from linear algebra of vector spaces. Urchin was tried to be refuted by indicating the null areas of the some polygons in it. However, mathematically speaking this metric approach to geometric objects is far from being a topological approach. As soon as the concept of area is introduced as a quantity, one might ignore, for instance, the degenerate polygons.

In other words, two objects might be homeomorphic but might have different areas.

For example, consider two cubes, one having area a, the other having area 2a. Clearly, these two cubes are homeomorphic, but their areas are different. Null areas appear when we have a dimension lacking such that some regions are still covered. It is essentially similar to the crossing lines problem which we discussed above. Immediately after this observation, the picture frame was handled by a new definition of polyhedra. Definition 5 In the case of a genuine polyhedron, through any arbitrary points in space, there is at least one plane whose cross-section with the polyhedron consists of a single polygon.

However, it had an Euler characteristic of 0.

The reason for this is that a torus has only one hole, or more properly speaking, a torus is of genus one. More precisely, one can always find a homeomorphism between a picture frame and a torus. Thus, the topological properties of a torus and a picture frame are identical.

Therefore, it would not be too strange to expect that the picture frame might be handled in some way via these properties. Definition 5 served for this purpose. It emphasized the simple connectedness requirement for Descartes-Euler polyhedra. Observe that if a point is taken in the picture frame, a plane whose cross-section with the picture frame consists of a single polygon does not exist. The following counterexamples challenge Definition 5. It was then claimed that the cylinder cannot be a counterexample to the definitions of a polyhedron, since it was inconsistent with the proper definition of an edge.

As a result, a new definition of an edge was proposed. Definition 6 An edge has two vertices.

We observe that by defining an edge, the cylinder was excluded from being a polyhedron by the methods of negative heuristics. Nevertheless, this is not the final step.

The consequent definition of an edge in this respect can be considered a product of positive heuristics as well. By subsequently discussing the definition of an edge, Lakatos demonstrated precisely how his MSRP worked.

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One negative heuristic can indeed be followed by a positive heuristic move in such a way that, similar to the Hegelian dialectics, a new concept would be generated. Let us now briefly review one such aspect, monster-barring. In monster-barring, we may theoretically redefine the terms recursively, i. Therefore, the process of definining becomes ad hoc, and we may need another tool to get ourselves out of this vicious circle of ad hocness.

This is precisely what Lakatos wanted us to do. This is because, it can be used to avoid the counterexamples or similarly to improve the conjecture by polishing the defining terms. Yet, this process, as we pointed out, is ad hoc.

Although it did exclude all the exceptions, we might not be able to determine whether this is true.

## PHILOSOPHICAL PAPERS

In fact, as was remarked in PR, the urchin and the cylinder are counterexamples to this new statement of the theorem. Instead, it focuses on the definitions, its domain and its truth sets—the objects that satisfy the conjecture. In other words, the defining terms of the monster urchin were redefined in such a way that the monster became a regular. It was then concluded that the urchin is a polyhedron and is Eulerian as well. However, its star-polyhedral interpretation was faulty.

In order to be able to utilize positive heuristics, it was then claimed that the urchin had triangular faces. Thus, this interpretation yielded the observation that the urchin was in fact Eulerian with the appropriate calculations. Yet, some objections were given against the monster-barring interpretation of the urchin. Since all of its triangles lie on the same plane in the groups of five and thus surround a regular pentagon behind the solid angle, it was claimed that the triangular interpretation was false.

In this case, these objections played the role of positive heuristics as all of the interpretations or misinterpretations exposed what the urchin really was and how it should have been truly understand.

Positive heuristics again helped us how and in which direction we should proceed. In this specific example, we were told how to analyse the urchin, and how to apply it to the Euler conjecture by these positive heuristics. The picture frame was considered first in this regard. It was concluded that the picture frame cannot be inflated into a sphere or a plane. The reason for this is that the genus of the sphere is zero, and this is true even under the planar interpretation of stretching.

In other words, a homeomorphism between these two manifolds exists. From this, it was concluded that the picture frame could be inflated into a torus. It should now be recalled that the torus has genus one. Hence, it can be concluded that the picture frame is not planar. This is the exact reason which made the picture frame a global counterexample. Since the torus is not simply connected, it was argued that the original conjecture holds only for simple polyhedra, namely for those which, once one of their faces has been removed can be stretched onto a plane.

However, the proof still remains the same. Apparently, by the introduction of the concepts of torus and that of genus, we were shown how to interpret the picture frame correctly. Thus, we observe that the positive heuristics introduced us to a new concept of genus in order to allow us to interpret the non-simple polyhedra precisely. The conjecture was improved and thus we ended up with a new formulation.

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Advanced Search. Article Navigation. Close mobile search navigation Article navigation. Volume XIV. Oxford Academic. Google Scholar. Cite Citation. Permissions Icon Permissions. Article PDF first page preview. Issue Section:.This is the exact reason which made the picture frame a global counterexample. Definition 1 A polyhedron is a solid whose surface consists of polygonal faces.

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In this context, determining how the Lakatos method functions and which steps of the method work can provide us with important clues about the its validity for the teaching and learning of mathematics. To the critics that say such a textbook would be too long, he replies: This guilty lemma may have previously remained "hidden" or may have been misidentified.

Geometric conjectures: The importance of counterexamples. Therefore, it is observed that there are many similarities between the environment proposed by NCTM standards and the environment prepared according to Lakatos's model.

Related articles in Google Scholar. Ma explained that some teachers had the same understanding about the perimeter and area of a figure.