Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs


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My previous work on [ 14 ] broke new ground in automated reasoning and diagrammatic reasoning. Hence, the time is ripe to look into combining the existing diagrammatic reasoning methods with the symbolic ones in a single framework which will still guarantee to construct a correct formal proof of a theorem - this is what I aim to do in the proposed project. Furthermore, my objective is also to enable a system to learn from examples of similar reasoning patterns - the existing research on this topic is described next.

Some work has been done in the past on applying machine learning techniques to theorem proving, in particular on improving the proof search [ 9 , 32 ]. However, not much work has concentrated on high level learning of structures of proofs and extending the reasoning primitives within an automated reasoning system. Silver [ 34 ] and Desimone [ 8 ] used precondition analysis to learn new method schemas, which are similar to our proof methods. Their systems require a lot of reasoning with one example to reconstruct the features which can then be used to prove a new example.

The reconstruction effort needs to be spent in every new example for which the old proof is to be reused. I have developed [ 21 , 20 ] an approach to learning semi-automatically a symbolically expressed proof method. In this proposed project I will need to investigate whether this learning approach can be applied to learning diagrammatic and heterogeneous proof methods. To summarise, in my past research I explored reasoning with diagrams [ 14 , 13 , 18 ], but not by applying proof planning techniques to proof search, and not within an environment that allows for construction of heterogeneous proofs.


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I also devised a basic framework for learning symbolic proof methods from examples of proofs [ 21 , 20 ], but not in diagrammatic reasoning. These encouraging preliminary results provide me with the right system, framework and techniques to form the basis for the proposed project. Aims and objectives The aims of this proposed project are to: Integrate diagrammatic reasoning within a proof planning environment. Combine diagrammatic and symbolic reasoning in a heterogeneous reasoning framework. Devise a learning framework for learning diagrammatic and heterogeneous proof plans.

Experiment with the system in order to try and discover new diagrammatic proof plans. Gain insight into understanding of human informal reasoning e. This project is novel, ambitious and open ended, but with a clear starting point. It addresses issues, and uses techniques and results from several fields of artificial intelligence including automated reasoning, machine learning and cognitive science. At the same time it requires and develops specialised knowledge of diagrammatic reasoning, proof planning, theorem proving and machine learning.

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In addition to the theoretical results about the nature of human informal reasoning in mathematics, the result of the project will be a semi- or fully-automatic reasoning system which can prove theorems diagrammatically, learn new proof plans, and hopefully discover new diagrammatic proofs of mathematical theorems.

The success of the project will be evaluated by measuring the extent by which the aims of the project listed above have been achieved. These will be achieved by realising the plan that I describe next - the research can be divided into five main work-packages. The first three are fairly clearly defined, hence, it is possible to estimate the time that they will take to achieve. The last two are more open-ended, and hence the time that individual tasks will take are more difficult to predict - the research will show the more precise direction that I will take.

A starting point for this project is to choose a problem domain with a wealth of theorems that can be represented and proved diagrammatically. For instance, in my past work I considered problems of discrete space i. A potential additional domain are problems of continuous space i. A set of possible diagrams and diagrammatic operations needs to be identified next. Clearly, I will not be able to study all possible types of diagrams, but some candidates are diagrams of discrete space see Figure 1 and continuous geometric diagrams see Figure 2.

I should define a mapping relation between these diagrams and sentential formulae e. The diagrammatic operations need to be represented as proof methods so proof plans can be formed from them. To begin with, only basic operations are required, such as combining rows and columns. More complex operations will be learnt by the system automatically. The next step in formalising the proposed reasoning framework is to construct examples of diagrammatic proofs from which new proof methods will be learnt.

The conjectures can be input either as sententially represented theorems, by the user, or interesting combinations of diagrams can be constructed automatically by the system see work-package IV. A mapping relation is used to map the sententially represented theorems into diagrammatic representation.

Next, the available diagrammatic proof methods are applied in order to construct proof plans.

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Initially, these can be input by the user. Later, their selection will be guided by the existing proof planning techniques. Tasks: I. Some proofs cannot be expressed entirely diagrammatically for a number of reasons. For example, it is difficult to express quantification, negation, disjunction, or taking square roots see the inequality theorem given earlier.

Theorems whose proofs consist of such features need to be identified e. Next, we have to consider how diagrammatic proof steps can be combined with symbolic ones in a uniform proof system. The proposed system needs to have a guarantee that such heterogeneous proofs are correct. Both, diagrammatic and symbolic, proof methods should be treated in the same way by the proof planning engine. Tasks: II. Learning new diagrammatic methods is important for the reasoner's ability to prove theorems.

For instance, consider again the theorem about the sum of odd naturals in Figure 1. If an lcut was not available, then the system would not be able to prove this theorem diagrammatically. Thus, this proof method needs to be made available to the user or the system which constructs the proof. The aim is that the proposed system will learn such a method automatically.

In this example, an lcut proof method could be learnt from examples of proofs that use the two basic operations, namely the cut of a row and a column which are joined to form an ell. In the development of the learning architecture I need to address the issue of how diagrammatic proof methods can be represented so that the learning is made easier.

That is, how much information needs to be abstracted from the method representation so that the learning techniques can be applied, and how this information can be restored. Furthermore, the learning algorithm that will be required in this project needs to meet at least the following requirements: it should be able to deal with different styles of reasoning e.

The learning approach described here aims to learn new higher-level proof methods on the basis of the already given ones. It cannot learn language extensions, i. This is to avoid mistaken " theorems ", based on fallible intuitions, of which many instances have occurred in the history of the subject. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics.

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Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may be erroneous if the used computer program is erroneous.

Axioms in traditional thought were "self-evident truths", but that conception is problematic. Nonetheless mathematics is often imagined to be as far as its formal content nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory. Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change i.

In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic , to set theory foundations , to the empirical mathematics of the various sciences applied mathematics , and more recently to the rigorous study of uncertainty. While some areas might seem unrelated, the Langlands program has found connections between areas previously thought unconnected, such as Galois groups , Riemann surfaces and number theory.

Discrete mathematics conventionally groups together the fields of mathematics which study mathematical structures that are fundamentally discrete rather than continuous. In order to clarify the foundations of mathematics , the fields of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects.

Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs - CRC Press Book

Category theory , which deals in an abstract way with mathematical structures and relationships between them, is still in development. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately to The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer—Hilbert controversy.

Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. Therefore, no formal system is a complete axiomatization of full number theory. Modern logic is divided into recursion theory , model theory , and proof theory , and is closely linked to theoretical computer science , [ citation needed ] as well as to category theory. In the context of recursion theory, the impossibility of a full axiomatization of number theory can also be formally demonstrated as a consequence of the MRDP theorem.

Theoretical computer science includes computability theory , computational complexity theory , and information theory. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with the rapid advancement of computer hardware.

The study of quantity starts with numbers, first the familiar natural numbers and integers "whole numbers" and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory , from which come such popular results as Fermat's Last Theorem.

The twin prime conjecture and Goldbach's conjecture are two unsolved problems in number theory.

Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs
Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs
Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs
Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs
Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs
Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs
Mathematical Reasoning: Patterns, Problems, Conjectures, and Proofs

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