Mathematical LogicIn 1935 a course by the Cambridge mathematician M. H. A. (Max) Newman introduced Alan Turing to the frontier of research in mathematical logic.
The twentieth century dawn
Already in 1933, Turing was familiar with the work of Bertrand Russell on the foundations of mathematics.
History of Set Theory
Text of Russell's The Principles of Mathematics (1903)
What Gödel saw
Kurt Gödel's 1931 incompleteness theorem rewrote the agenda in the foundations of mathematics.
Gödel's 1931 work left open the question of the decidability of mathematical propositions, and this is what Turing set out to answer. The particular technique of Gödel numbering was also influential in Turing's 1936 work. Gödel had shown how to encode theorems about numbers, as numbers. Turing went on to show how to encode operations on numbers, by numbers.
Hilbert and decidability
The famous 1900 speech by the German mathematician David Hilbert did much to set the agenda for twentieth century mathematical research. More comment by Ivor Grattan-Guinness.
Hilbert's 1900 Tenth Question about the solubility of Diophantine equations was a precursor to his later question about the 'decidability' of (apparently) more general mathematical assertions, the Entscheidungsproblem (decision problem) which inspired Turing's work.
Martin Davis's 1958 text Computability and Unsolvability did much to propagate Turing's work and later editions of his book added Davis's beautiful exposition of the resolution of Hilbert's Tenth Problem in 1970, in which his own work had been an important ingredient. Martin Davis's later book The Universal Computer, the road from Leibniz to Turing gives a fascinating introduction to the development and significance of mathematical logic, culminating in a detailed account of Turing's work.
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A turing machine can be defined as a “device with a finite number of internal configurations, each of which involves the machine’s being in one of a finite number of states”. This means that in a machine there are certain inputs that when implemented will produce a certain output. These combinations of inputs can be used in a machine table to show the outputs for each state. By looking at a machine table one will be able to know which combination of inputs is related to the different states and also what happens at each state.
Mental states can correlate directly with turing machines in that our minds can be described by using a machine table. The inputs for the machine table representation of our mental states can be seen as our perceptions. By saying perception I mean all the sensory body parts/organs that can perceive an input and send a signal to the mind. These inputs in turn determine the mental state and the behaviors related to each mental state. Because our minds can be mapped out using a machine table, each state can be seen as an instance of functionalism. Functionalism is defined as an object that has a function. Functionalism can be used to describe a single mental state.
An example of a mental state that is determined by a turing machine can be the feeling of happiness. In order to be happy there are certain things that are needed to trigger it. This mental state of being happy is triggered by our perceptions and then causes one to express their mental state physically. Our physical expression, I believe, can be altered. For example, if one were told that a dog was called a cat and only knew that dogs were cats then he would always see a dog and call it a cat. It is only because of what we hear and learn from our environment that we call a dog a dog.
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Turing Machine Computer Engineering Functionalism Table Combination Feeling Organs Input Signal
123helpme.com/search.asp?text=computer+engineering">computer engineering major and after taking a course in circuit design last quarter I found a resemblance with the state maps or truth tables that we used in circuit design and the machine tables described by Putnam. Both tables consist of inputs and states. The states determine what the chip will physically do and the state is determined by the inputs. This means that circuits can be a representation of a turing machine and could be an explanation for the mind as a computer.