Computers, Maths and Minds _ Plus.maths

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Figure 1: The idea of a function.

Submitted by mf344 on February

Many of us own computers, and we (well, most of us) have a rough idea that computers are made up of complicated hardware

software. But perhaps few of us know that the concept of a computer was envisioned long before these machines became

ubiquitous items in our homes, offices and even pockets. And as we will see later, some have even suggested that our own br

are embodiments of this theoretical concept.

The first electronic computer was built during the second world war at Bletchley Park, as a device to crack the German Enigma

(see here (http://plus.maths.org/content/exploring-enigma) to find out more). Instrumental in its construction was Alan Turing

(http://plus.maths.org/content/alan-turing-ahead-his-time) , who in 1936 had developed the theoretical machine mentioned above. To

understand it, let's start with the underlying mathematical concept of a function. Mathematical functions are roughly like mach

in that they take in some data and churn out some information. (There are many different metaphors on functions, but let's s

to this one.) They are sometimes also called mappingsor transformations. An example is the function f(n) = 2n: given an inp

value for n, say n = 2, the function returns the value f(2) = 2 x 2 = 4.

In order to specify a function it's not enough to come up with a formula, or other type

of rule, like the one above. You also need to specify the domainand co-domain. Thedata, or input, we feed this machine, has to come from a set of elements; that's the

domain. On the other hand, the information, or output, churned out by this machine

belongs to another set of elements; that's the co-domain. Figure 1 illustrates these

ideas. In our example above, if the domain consists of all the naturalnumbers, then the

co-domain consists of all even natural numbers. If we specified the domain to consist of

all real numbers on the number line, then the co-domain would also consist of all real

numbers.

Functions play a fundamental role in mathematics, but pragmatists who wish to apply

these wonderful ideas to build technological devices need to find ways to implement

them in machines. A very good example of this is the computer, which processes

instructions using algorithms. To paraphrase the Berkeley computer scientist Christos

Papadimitriou, algorithms are "precise, unambiguous, mechanical, efficient, correct"

procedures to generate outputsgiven inputs. In short, they are the means by which a

function is executed on a machine.

What happens, though, if we try to conceptualise this entire procedure? Could we imagine an abstract entity that describes th

whole "computer-algorithm-input-output" jazz?

Turing machines

This is exactly what Turing did. He proposed a theoretical model of a computer, now called the Turing machine. A simplified ve

of the machine looks like this:

Computers, maths and mindsby Alan Aw (/content/list-by-author/Alan Aw)
http://plus.maths.org/content/alan-turing-ahead-his-timehttp://plus.maths.org/content/http://plus.maths.org/content/http://plus.maths.org/content/http://plus.maths.org/content/list-by-author/Alan%20Awhttp://plus.maths.org/content/alan-turing-ahead-his-timehttp://plus.maths.org/content/exploring-enigmahttp://plus.maths.org/content/


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The zeros and ones appearing on the infinitely extensible tape are called symbolsand form the alphabetthe machine is using.

the general version, the alphabet has symbols, where is a natural number, but to make things easier, you could think of tonly being two symbols, 0 and 1. At any one time the machine exists in one of possible states . The machin

a reader, which reads off one symbol of the tape each time. And the reaction to that reading is determined by the state the

machine is in.

After reading off a symbol, the reader can either leave it as it is, or erase it and replace it with a new one. It then moves either

step to left or to the right and reads the symbol there, or stops altogether. Which one of these actions it takes, and what new

symbol it writes if indeed it does, depends on the state it's in. Concurrently with these actions its state may also change. Whe

then meets the next symbol on the left or right, it reacts according to the state it is now in. The transitory instructions that t

machine what to do at each step are collectively termed the machine'sprogram .

An interesting way to capture a Turing machines program is to use quintuples of symbols. For example, the quintuple

would signify "if you are reading a 0 and are in state , then replace the 0 by a 1, move one step to the right and change your

state to " Another quintuple now dictates what the machine is to do when it finds the symbol 1 after moving to the right. F

example,

tells the machine to leave the 1 as is, move another step to the right, and change to state

Given an input a tape full of symbols the Turing machine generates an output. And by programming it correctly, you can

to perform all sorts of calculations. An example is the addition of positive natural numbers. If you want to add the two numbe

and 4, prepare your input tape with 1s in the first two positions, then a 0 in the next position along, followed by four 1s in th

following four positions, and set all the other positions to 0. So you have

110111100000.....

The rules for the machine to follow are:

State 0: If current position has a 1, move one step to the right;

State 0: If current position has a 0, replace it with a 1, move one cell to the right and change state to state 1;

State 1: If current position has a 1, move one cell to the right;

State 1: If current position has a 0, move one cell to left and change state to state 2;

State 2: If current position has a 1, replace by a 0 and stop.

By following these rules, the machine changes the 0 position inbetween the 1s to a 1 and goes to the end of the string of 1s

erases the last one. In this way, we get six 1s in a row, representing the desired result.

It could of course happen that a program results in the machine keeping on going forever, for example by entering an infinite

The problem of determining whether a given program will ever stop led to some very interesting developments in mathematica

in its own right, see this article (http://plus.maths.org/content/omega-and-why-maths-has-no-toes) . But let's continue with our notion of

functions.
http://plus.maths.org/content/omega-and-why-maths-has-no-toes


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Alan Turing

How does our brain work?

Church, Turing and the Church-Turing thesis

Suppose we are given some mathematical function with the set of natural numbers as its domain, and a Turing machine is a

generate the output for each input . Then lets call Turing computable. Another way to see this is that the Turing mac

model "works" for our particular function .

The obvious question now is which functions are Turing computable and can therefore be

implemented on a Turing machine. If the function assigns outputs to inputs in a totally random

without rhyme or reason, we can't really hope to get it to work on a Turing machine, so let's

restrict our attention to functions that assign an output to an input using a finite sequence of

algorithms (i.e., a finite list of simple and mechanical instructions) such functions are called

computable.

Combining ideas of Turing computable functions and generally computable functions, we may e

convince ourselves that since the Turing machine seems to only involve three steps (moving le

moving right, replacing symbols, or halting) it is mechanical enough and so any function that is

Turing computable is also computable. Likewise, since the Turing machine works in a mechanica

fashion, perhaps any algorithm for computing a function is expressible as a program on a Tur

machine; that is, computable functions are also Turing computable if this is true, then any

reasonably mechanical calculation can be implemented on a Turing machine.

The idea that computable functions are computable on a Turing machine is called the Church-T

thesis, acknowledging the work of the American mathematician Alonzo Church (http://www-history.mcs.st-and.ac.uk/Biographies/Church

Alas, no one has been able to verify this seemingly true observation. This is partly because it's quite hard to pin down what ex

we mean by a more general computable function.

Brains as computers?

Assuming that the Church-Turing thesis holds (many researchers do anyway, and some have even considered using it as a

definition!), it follows that all computers Windows, iMac, you name it are effectively modelled by Turing machines. Making

even greater intellectual leap, if we see the brain or mind as effectively a supercomputer of sorts, then what follows is that ouris a Turing machine.

Some have suggested that there is increasing evidence for the truth of this. I recently

attended a sharing session by Shaun Martin, a former mathematician, who introduced

me to the "cognitive revolution" the coming together of many disciplines including

psychology, biology, philosophy, linguistics, anthropology and computer science, with

the aim of understanding human cognition. The revolution provides pristine

perspectives on the centuries-old question of "What is the mind"? (Interested readers

might wish to visit this (http://en.wikipedia.org/wiki/Philosophy_of_mind#Cognitive_science)

Wikipedia article.) And as we have just seen, there is a mathematical side to this

endeavour as well.

As Martin pointed out, modern research from multiple disciplines, most notably

psychology and experimental philosophy, seems to suggest a sort of underlying

universal framework upon which every human mind is based. For example, experiments

suggest that our brains are wired to respond in two different manners: quickly and

intuitively, versus slowly and rationally. See the further reading list below for some

books and articles that support this idea. If our responses are hard-wired, then

presumably they could be mimicked by a sufficiently sophisticated computer.

If there is so much empirical data supporting this vantage point, then perhaps

unsurprisingly there are also detractors who argue that the mind is more than a set of

algorithms or functions. In the 1980s the American philosopher John Searle

(http://en.wikipedia.org/wiki/John_Searle) imagined a scenario, now known as the Chinese room thought experiment

(http://en.wikipedia.org/wiki/Chinese_room) , in which a computer runs a set of instructions in Chinese translation so well so that it is

perceived by passers-by as a real human being. Regardless, for Searle this does not mean that the computer possesses a min

consciousness in the sense that humans do. That is, there is a fundamental distinction between the human mind and merecomputer programs. (More arguments may be found in the articles listed in the further reading list below.)

We hitherto do not know enough to decide which theories we ought to perceive as correct. (I definitely don't.) Shaun Martin

grinned delightfully at us when he talked about the mind as a Turing machine. Clearly, his beautiful mathematical insight into th

multidisciplinary venture is that perhaps, while maybe the mind isn't all about algorithms, at least some part of it could be (the

universal aspects as far as research reveals them to exist). And so it boils down to figuring out what these fundamental algori

are that govern our existence, and what implications their discovery might have towards bigger questions concerning humanit

Further reading

Read more about Turing machines on Plus (http://plus.maths.org/content/taxonomy/term/230)

The blank slate (http://www.amazon.co.uk/Blank-Slate-Modern-Penguin-Science/dp/014027605X/ref=sr_1_1?s=books&ie=UTF8&qid=1390820263

1&keywords=The+blank+slate) by Steven Pinker

Thinking, fast and slow (http://www.amazon.co.uk/s/ref=nb_sb_ss_i_1_20?url=search-alias%3Dstripbooks&field-
http://www.amazon.co.uk/s/ref=nb_sb_ss_i_1_20?url=search-alias%3Dstripbooks&field-keywords=thinking+fast+and+slow+by+daniel+kahneman&sprefix=Thinking%2C+fast+and+s%2Cstripbooks%2C730http://www.amazon.co.uk/s/ref=nb_sb_ss_i_1_20?url=search-alias%3Dstripbooks&field-keywords=thinking+fast+and+slow+by+daniel+kahneman&sprefix=Thinking%2C+fast+and+s%2Cstripbooks%2C730http://www.amazon.co.uk/Blank-Slate-Modern-Penguin-Science/dp/014027605X/ref=sr_1_1?s=books&ie=UTF8&qid=1390820263&sr=1-1&keywords=The+blank+slatehttp://plus.maths.org/content/taxonomy/term/230http://en.wikipedia.org/wiki/Chinese_roomhttp://en.wikipedia.org/wiki/John_Searlehttp://en.wikipedia.org/wiki/Philosophy_of_mind#Cognitive_sciencehttp://www-history.mcs.st-and.ac.uk/Biographies/Church.html


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keywords=thinking+fast+and+slow+by+daniel+kahneman&sprefix=Thinking%2C+fast+and+s%2Cstripbooks%2C730) by Daniel Kahneman

Zoon Politikon: The evolutionary roots of human sociopolitical systems (http://www.umass.edu/preferen/gintis/StrungmanForum.pdf

Herbert Gintis and Carel van Schaik

The fundamental distinction between brains and Turing machines

(http://web.mit.edu/asf/www/PopularScience/Friedman_BrainsAndTuringMachines_2002.pdf) by Andrew Friedman

Morality is a culturally conditioned response (http://philosophynow.org/issues/82/Morality_is_a_Culturally_Conditioned_Response) by Jess

Prinz

About the author

Alan Aw is a maths enthusiast from Singapore who also loves running, reading, and thinking. He

served in the Singapore army and will be studying at Stanford University in 2014.
http://www.amazon.co.uk/s/ref=nb_sb_ss_i_1_20?url=search-alias%3Dstripbooks&field-keywords=thinking+fast+and+slow+by+daniel+kahneman&sprefix=Thinking%2C+fast+and+s%2Cstripbooks%2C730http://philosophynow.org/issues/82/Morality_is_a_Culturally_Conditioned_Responsehttp://web.mit.edu/asf/www/PopularScience/Friedman_BrainsAndTuringMachines_2002.pdfhttp://www.umass.edu/preferen/gintis/StrungmanForum.pdfhttp://www.amazon.co.uk/s/ref=nb_sb_ss_i_1_20?url=search-alias%3Dstripbooks&field-keywords=thinking+fast+and+slow+by+daniel+kahneman&sprefix=Thinking%2C+fast+and+s%2Cstripbooks%2C730

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