AND AI

Angel Garrido

Departamento de Matemáticas Fundamentales, Facultad de Ciencias de la UNED, Paseo Senda del Rey 9. 28040-Madrid, Spain; E-Mail: [email protected]

Abstract: From ancient times, the history of human beings has developed by a succession of steps and sometimes jumps, until reaching at the relative sophistication of the modern brain and culture. Researchers are attempting to create systems that mimic human thinking, understand speech, or beat the best human chess player. Understanding the mechanisms of intelligence, and creating intelligent artefacts are the twin goals of (AI). Great mathematical minds have played a key role in AI in recent years; to name only a few, Janos Neumann (also known as ), Konrad Zuse, Norbert Wiener, Claude E. Shannon, Alan M. Turing, Grigore Moisil, Lofti A. Zadeh, Ronald R. Yager, Michio Sugeno, Solomon Marcus, or Lászlo A. Barabási. Introducing the study of AI is not merely useful because of its capability for solving difficult problems, but also because of its Mathematical nature. It prepares us to understand the current world, enabling us to act on the challenges of the future.

Introduction

The Mind and Brain can be thought of as computational systems — but what kinds of computations do they carry out, and what kinds of mathematics can best characterize these computations?

The historical origin of Artificial Intelligence as a scientific field is usually established at the Darmouth Conference (1956). In that year, John McCarthy coined the term, defined as “the science and engineering of making intelligent machines”. But this definition does not cover the actual breadth of the field. AI is multi-connected with many other fields, such as Neuroscience or Philosophy; and we can trace its origins further back, to arcane origins, perhaps to Plato, Raymond Lully (Raimundo Lulio, in Spanish), G. W. Leibniz, Blaise Pascal, Charles Babbage, Leonardo Torres Quevedo, etc., with their attempts to create thinking machines. A suggestive definition of the field would be to say that it is the area of Computer Science that focuses on creating machines that can engage on behaviours that humans consider intelligent.

The history of AI field is long and fruitful; just recall topics as diverse and important as the famous Turing test, the Strong Vs Weak AI discussion, the Chinese Room argument, and so on. And modern AI has spanned more than fifty years now.

Frequently, AI requires Logic. But Classical Logics show too many insufficiencies [7, 8], that have made necessary to introduce more sophisticated tools, such as Fuzzy Logic, Modal Logic, Non- Monotonic Logic. Indeed, Mathematics can be thought of as a mere instance of First-Order Predicate

Calculus, and therefore, a part of applied Monotonic Logic. The limitations of classical logic reasoning, and the clear advantages of Fuzzy Logic, are discussed later in this paper.

Among the things that AI needs to implement as representation are Categories, Objects, Properties, Relations, and so on. And all of these are connected to Mathematics [1-6]; they are, as well, very good and illustrative examples, in the context of education. For instance, it is possible to show Fuzzy Sets together with the usual sets, also called Crisp, or Classical Sets, as a particular case; or to introduce concepts and strategies from Discrete Mathematics, as the convenience of using Graph Theory tools on many fields.

The problems in AI can be classified in two general types: Search Problems and Representation Problems. Then, we have Logics, Rules, Frames, Nets, as interconnected models and tools, by graphs. As it is easy to see, all of them are very mathematical topics.

The origin of ideas about thinking machines [2, 7], and the mechanisms through which the human brain works; the possibility of mimicking its behaviour if we create some computational structure similar to the neuron, or perhaps the neural system, its synapses or connections between neurons, to produce what is called a Neural Network… Rather than Science Fiction, or the plot of a film, these are real-world subjects of study, and they have been so for many years. And the interest in them has increased the near past.

The central purpose of AI would be to create an admissible model for human knowledge. Its subject is, therefore, “pure form”. We try to emulate the way of reasoning of a human brain. Research directed to this goal can only happen in a succession of approximating steps, but the attempts proceed always in this sense.

Initially, AI worked over idealizations of the real world. Its natural fields were, therefore, “formal worlds”. Search procedures operated in the Space of States, which contains the set of all states (or nodes, in the case of representation by graphs), that we can obtain when we apply all the available operators. Many early AI programs used the same basic algorithm. To achieve some goal (winning a game or proving a theorem), they proceeded step by step towards it (by making each time a move or a deduction) as if searching through a maze, backtracking whenever they reached a dead end. This paradigm was called “reasoning as search”.

Studying Artificial Intelligence is interesting not only because its potential to tackle many open problems, both inside the field and in application to others scientific areas, and even the study of the humanities, but also because it is a new and very creative branch of Mathematics, and it prepares us to understand the current world, enabling us to act on the challenges of the future.

Ray Kurzweil (RK, by acronym) is one of the most brilliant and active current minds, in the field of AI. This is an inventor and a soothsayer, or diviner of the future of technology, on which has made many predictions (see, for instance, his book The Age of Spiritual Machines), most of which have been met. It becomes a kind of Jules Verne´s reported.

Among his most notable works include:

- . It refers to time (the `singularity´) in which the machines will reach and surpass the power of human thought: he figure in 1945.

- How to Create a Mind? On the functioning of the brain (especially the structure of the neocortex), and how it could imitate or simulate reach into the computer.

In 1990 he predicted that a computer could defeat a world chess champion in 1998, and a year before coming to that date, as did the DEEP BLUE, IBM, playing against Garry Kasparov; or defeat a world champion in GO, more complex play, by Deep Mind. Also predicted the of the WWW, previously only used by some people in `academia´. This and his other predictions have been performing like that in 2000 would be available as of prosthetic limbs that allow paraplegics walk, which is already used, for example, US Army veterans. Also the appearance of `automatic´ driver able to bring himself automobiles. And getting .

Among his many inventions, we could mention the first electronic reader for the blind people, or system which resulted in the . Also the first digital scanner. It has created business and has sold and then created others. Currently, RK is head of research at Google, in San Francisco.

RK said, in his book How to Create a Mind?, “There is now a grand project under way involving many thousands of scientists and engineers working to understand the best example we have of an intelligent process: the human brain. It is arguably the most important effort in the history of human- machine civilization”… “reverse-engineering the human brain may be regarded as the most important project… The goal … is understand precisely how the human brain works, and then to use these revealed methods to better understand ourselves, to fix the brain when needed and … to create even more intelligent machines. Keep in mind that greatly amplifying a natural phenomenon is precisely what engineering is capable of doing. As an example, consider the rather subtle phenomenon of Bernoulli´s Principle, which states that there is slightly air pressure over a moving curved surface than over a moving flat one. The mathematics of how Bernoulli´s Principle produces wing lift is still not yet fully settled among scientists, yet engineering has taken this delicate insight, focused its powers, and created the entire world of aviation.

His project is to come to understand what human language means. Because when you write an article you are not creating a collection of interesting words. Is that you have something to say, and Google is being devoted to intelligently organize and process all the information. The message of our paper is information. We want the computer to read everything available on the Web, and all pages of each of the existing books, to become able to maintain an intelligent dialogue with the user, as well as to be able to answer your questions... Google will know the answer to your question even before you ask it...

In a recent interview (2015) RK was asked by the difference between a human brain and a computer. He replied that “a computer is not able to do two things at once. Just last, but does it very quickly and very accurately. Make billions of operations per second, and just wrong. The brain is a bit contrary. Just make a few operations per second; this makes it slower than the computer. “And when

will this be possible?”, they asked. “We have no long said to reach that point. I wrote it would take a computer capable of performing 100 trillion and 100,000 trillion operations per second for `recreate´ the human brain. Today there are `giant´ computers with that ability. I think that in just a decade there for a thousand dollars.” And then he was asked: “¿As soon? And replied that “My thought and comments were even quite conservative, that at the end of the decade will be very cheap computers with sufficient capacity as to simulate the human brain. Which is already possible with the means which currently available to Google.

RK features that are there creating a model capable of handling the language the same as humans, that is, not only to read documents, but to understand them. A system capable of reading billions of pages, so you learn anything of interest to appear on the Internet, creating a KB (Knowledge Base) about the world, which we can go with questions. “And what is more difficult to simulate in the human brain?” they asked. “The level of abstraction that is capable of achieving”, he replied. “I think the irony or sarcasm [or polysemy in general]. Because they involve several levels of interpretation, and this confuses computers. What makes the human brain is creating a hierarchy with respect to their own thinking. Therein lies the secret of how we have any ideas on how to create it on the computer: a process that can be interpreted, and replicate too.

“Can a computer to compose a symphony or write a novel?”- asked too. He said “these are the most complex creations of our neocortex, as they work through metaphors. Each type of music is a language that few people come to dominate. Only a handful of geniuses reach the level of a Mozart or a JS Bach. Among other things, because our brain has a very limited capacity: just 300 million neural modules. But soon we will to improve our brain, and have some kind of additional, external neocortex, which is available in the `cloud´, allowing us to overcome this limitation. You need to have a special brain to become a genius like Mozart, and still do not fully understand the processes involved. But skills are less important than what have come to be so far because we will not have such a limited brain, since our neocortex may be much more powerful. We can connect you to the outer brain around 2030, and will also be a flexible body”.

Can be considered RK as a clear representative of the so-called `hard AI´, as may be Marvin Minsky? Perhaps, but some others defend the `soft AI´, questioning him go to get. So, the British cosmologist , in his book The Emperor´s New Mind, ensures that a computer can never have intuition or imagination. Penrose argues that human is non-algorithmic, and thus is not capable of being modelled by a conventional Turing machine-type of digital computer. Penrose hypothesizes that plays a very essential role in the understanding of human consciousness. The collapse of the quantum wave function is seen as playing an important role in brain function.

Among his many inventions (of RK), we could mention the first electronic reader for the blind, or speech recognition system which resulted in the SIRI. Also may be mentioned the first digital scanner. It has created business and has sold and then created others. Currently, he is head of research at Google, in San Francisco.

On what obstacles would have to come to simulate its operation by means of a computer, he replied that “many, but not impossible. The British mathematician Alan M. Turing said that any form of computing in the end, is equivalent. So a device with sufficient memory could simulate the operation of any other device, which would also apply to the computer. Since it may simulate the behavior of the human brain, by performing the same operations just very quickly.

“An artificial human brain can be built within the next ten years”, said Henry Markram, Israeli- born scientist, professor at the Polytechnic of Lausanne and director of the (BBP), also called Human Brain Project. This is a project by the Swiss government, and also funded by private donations, with the aim of trying to build an artificial copy of the mammalian (and therefore human) brain. In 2013 it was assigned to this project (and another on graphene) many euros for research over the next ten years. Turning to this Project, the team of Prof. Markram is concentrated in the study of the `neocortex´, which it is the layer of the brain responsible for higher functions directly, as would (for example) the conscious thought. In recent years, this team has managed to decipher the structure of such `neocortical column´. Working with a software model that reproduces tens of thousands of neurons, all different from each other, which helped them get to artificially help build that column. All this opens up new and promising perspectives for science in its support for improving the lives of humans. We would need a laptop for each neuron. Therefore, ten thousand laptops need. Each neuron would require one of those laptops. So this team is to use an IBM supercomputer with ten thousand processors.

Through simulations we can observe brain function and behavior in dysfunctional situations, as in the case of depression or Alzheimer´s. Researchers may prove what is the response of new drugs on these dysfunctional patterns. “The tool will be able to understand what happens in the brain and may try different behaviors. It's like in aeronautics: a model simulates the flight and can test how it would affect the wind, rain and other weather conditions, and how they respond to the aircraft. Are trying to do something similar with the brain”, exemplifies Prof. Peña. It is expected that in the future, neuroscientists know how is formed, develops and ages the brain, or the mechanisms by which we learn and improve our intellectual capacities.

We should also mention another program, somewhat parallel in purpose to the previous, which is the American BRAIN project, behind which are researchers as the Spanish Rafael Yuste, professor at Columbia University. Their budget is for a similar amount, but this time in dollars, and aims to emulate the achievements of the Human Genome Project. It can be seen as a kind of `spatial´ new career, but now between Europe and the USA. The map of the brain will be built thanks to these new `cartographers´ of neuroscience, an initiative that will see its results in the coming years. In these projects collaborated big companies like Google, either IBM, or Microsoft.

Quantum computing is the `next generation´ of technology to transform our world. It would be based on the properties of quantum mechanics completely different to those who taught us (were the so-called `Classic´). In the world of the smallest particles laws of nature that are accustomed to assume not valid. Now using particles like photons, has come to create an element: the minimal unity of quantum information, named Qubit (Quantum bit), which allows us to process information so that if

you dedicate centuries of classical computation, not make it. And as in classical is linear, in quantum case it can be solved more than one operation at a time.

Lately has been concentrated on other tasks, such as developing new recipes from existing. Trying to create a mathematical model that allows us to understand cultural evolution is still a very difficult task, but nevertheless, is beginning to be addressed by authors such as Donald Davidson, Daniel Dennett, as well as some theorist of artificial intelligence. We are referring, of course, the `memes´ and transmission (a concept introduced by Richard Dawkins in The Selfish Gene), using an analogy with genes and the evolutionary process. The fundamental thesis of Dawkins, on this issue, is that cultural traits, or `memes´, also replicated. But whereas the chromosomes are independent of our actions natural unit and the cultural dimensions would, however, our own `constructs´. It is a matter then of growing interest. Also it must be completed with related problems by recent publications as [11-14].

Keywords: Mathematical Education; Mathematical Logic; Fuzzy Logic; AI; Cybernetics; Computer Science.

REFERENCES

1. Courant, R. What Is Mathematics? An Elementary Approach to Ideas and Methods. OUP, 1996.

2. Mc Corduck, P. Machines Who Think (2nd ed.), Natick, MA: A. K. Peters Ltd., 2004.

3. Polya, G. How to Solve It: A New Aspect of Mathematical Method. Ishi Press, 2009.

4. Ibid., Mathematics and Plausible Reasoning. Vol. I: Induction and Analogy in Mathematics. Vol. II: Patterns of Plausible Inference. Both in Princeton University Press, 1990.

5. Ibid., Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving. Wiley, 1981.

6. Ibid., and Kirkpatrick, H. The Stanford Mathematics Problem Book: With Hints and Solutions. Dover Mathematical Books, 2009.

7. Russell, S., and Norvig, P. Artificial Intelligence: A Modern Approach (2nd ed.), Prentice Hall, 2003.

8. Searle, J., “Minds, , and Programs”, Behavioral and Brain Sciences 1980, 3 (3): 417–457.

9. Taylor, H. and L., George Pólya: Master of Discovery. Book Surge Publishing, 2006.

10. Turing, A., “Computing Machinery and Intelligence”, Mind 1950, LIX (236): 433–460.

11. Garrido, A., Lógicas de nuestro tiempo. Editorial Dykinson. Madrid, 2014.

12. Ibid., Lógica Aplicada. Vaguedad e Incertidumbre. Editorial Dykinson. Madrid, 2015.

13. Ibid., Lógica Matemática e Inteligencia Artificial. Editorial Dykinson. Madrid, 2016.

14. Ibid., Filosofía y Computación. Editorial Dykinson. Madrid, 2017.