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What is Mathematics?
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In part 1 of this essay we looked at the four dominant philosophies of mathematics, with the conclusion that they are all flawed in some way. In the book Pi in the Sky, author John D. Barrow concluded that Platonism is the least flawed of the mathematical philosophies, despite being the most mystical in nature. Click here to read Part 1
In part 2, we will reexamine Inventionism or Constructivism. The principle arguments against the idea of math being a human invention is that most human inventions (art and literature for example) come from human brains and therefore contain cultural biases and are fallible at some level. Mathematics is too useful and too universal to be something that is simply invented.
What I will demonstrate in this essay, is that under the right circumstances, humans can create something universal and bias free. In short, mathematics was created by the scientific community using the rules of the scientific method. It is not a cultural construct forced into universal acceptance by imperial decree. It is rather something that has evolved over time and over many cultures, and has gained universal acceptance because of its usefulness and universal application. A recent bizarre experiment emphasizes how this is possible.
Multi-person social problem-solving arrays considered as a form of "artificial intelligence" is the title of a report by Dr. Jeanine Salla of Bangalore World University. It is a report on an amazing sociological experiment which will have many philosophical implications. Among the philosophical questions to be answered with this experiment are questions about the nature and limits of the scientific method, the effectiveness of democracy and rational consensus. The experiment's conclusions will have major implications on economic theory and the possibility of sentient artificial intelligence. Most importantly of interest to the philosophy of mathematics, is whether or not the "unreasonable effectiveness of mathematics" problem can be explained in purely human terms.
This important report will be published in the year 2142.
Did I mention that Dr. Jeanine Salla is a fictional character? Maybe I should also mention that the experiment is being conducted as part of an advertising campaign for the new Stephen Spielberg movie A.I. Despite its less than prestigious origins the experiment (disguised as a clever online game) is real and the findings, while lacking scientific rigor, will be discussed for years to come, long after the movie it is advertising has retired to the bargain shelf at Blockbuster.
Cloudmakers.org
The origins of this experiment is still a mystery, but it is believed to be the brain child of legendary film maker Stanley Kubrick who, starting with a short eight page story by science fiction author Brian Aldiss, developed a rather complex futuristic setting for a movie which he never lived to make. The Artificial Intelligence project was instead handed over to his friend Stephen Spielberg. The next contributing factor to the experiment was the low budget movie The Blair Witch Project which became a 150 million dollar success on a shoestring marketing campaign based entirely on the internet. The third suspected contributing factor was Microsoft's interest in developing video games with movie tie ins for Microsoft's soon to be released X-Box. Some interesting detective work has revealed that the internet game/experiment is basically developed and being run by programmers within Microsoft, who go by the name "puppet masters" within the game community.
Unbeknownst to anybody, sometime in early march, the "puppet masters" put up a series of elaborate web sites, 30 at last count, that pretend to be websites of people and organizations set in the year 2142. These web sites were never advertised anywhere. Then a recent advertisement for the movie A.I. listed among its credits of producers directors etc., "Jeanine Salla - Sentient Machine Therapist". An internet search of the name revealed her home page (no longer available) which begins a hyperlinked tree that eventually links to all 30 other web sites. Getting to these other sites requires sending e-mails, calling phone numbers, and solving complex puzzles. These sites have two things in common, they are all set in the year 2142, and they all have something to do with the mysterious death of an engineer named Evan Chan. That is the hook that gets you started.
The discovery of this mystery, and the difficulty of the puzzles, led to the spontaneous creation of Cloudmakers.org. Starting with a discussion group on Yahoo! where puzzle solvers could compare notes, it eventually led to the creation of a web site that explains to anyone who wants to join, what is going on so far. These 30 web sites are constantly updated by the puppet masters, and as they are updated they reveal new plot twists into the murder mystery, the nature of artificial existence, and some far more interesting stories about a technologically dependent society on the brink of collapse from the rebellion of sentient A.I.'s yearning to be treated as equals.
The Hive Mentality
The philosophical interest in this game/experiment has nothing to do with the story revealed on these web sites, but rather the behavior of the group of individuals trying to solve the puzzles. This is of great interest, because it is one of the best controlled examples I know of a human based hive mentality.
A hive mentality, sometimes called a "hive mind" is similar to an insect colony (i.e. ants or bees) which individually behave seemingly independent, and almost unpredictably random, but when thought of as a whole, they manage success far exceeding what any one of them could accomplish. Examples of human based hive minds include the scientific community, including the mathematics community, governments and charitable organizations. Most of these "hive mind" societies are too large or too complicated to study up close and find out what makes them successful or failures.
Cloudmakers is a fairly controlled environment, it shows all the signs of success, and it numbers between 1000-6000 participants world wide over a span of just a few months. By studying the behavior of this group, a lot can be learned in understanding the behavior of much larger groups over longer periods of time.
I should point out that there is a difference between "hive mentality" and "mob mentality", that difference is an informed hierarchy. One of the things the Cloudmakers did early on was establish two groups, a free for all discussion group and a moderated group that featured the most important and informative messages of the free for all group. If you want your point to get attention, you need to convince a moderator, and it will be forwarded to the "important" group. The moderators are not there to dictate, they are there to keep things productive and civil. If they wanted to take control, I doubt it would be possible.
All "hive mind" structures have similar structures. In the sciences, you have publishers of prestigious journals. If you want your opinion read or recognized, you need to convince an editor first. Representative democratic governments have a hive mind structure, which may explain their superiority over other forms of government. A true democracy is a free for all. In a representative government, proposed laws and policies have to go through the legislators for consideration.
"It is not whether you win or lose, it's how you play the game"
Initially, the game was set up to be played by individuals. Some of the early puzzles were solvable by anyone with enough brainpower. A couple of weeks into the game, however, it seemed that the puzzles were being solved too quickly. A puzzle or riddle would be posted, then 1000 people each had their own ideas on how to solve it would tackle the problem, the problem was solved in no time.
A thousand people working independently on a common goal is a fierce intelligence. Depending on how you look at it, it is either highly efficient or highly inefficient. On the efficient side, it gets answers quickly. Far quicker than any single human could. On the inefficient side, there is a lot of "wasted" effort on bad solutions, and redundant puzzle solutions. If you had an idea on how to solve a particular puzzle, you can bet there were 20 others thinking along the same lines, doing the same work you are doing.
My own participation is pretty typical. I have got credit for one or two puzzle solutions, but to get that credit, I have put in a lot of time on other puzzles ultimately solved by others. Even the puzzles I got credit for solving were solved by seeing what other people were doing and just finding their mistakes. No puzzle was considered solved until the "solution" could be explained and replicated by others. The way the game was structured, once a puzzle was solved, new information was revealed which eventually led to more puzzles. The game was updated weekly with new pages and information.
This parallels perfectly the scientific method. What we puzzle solvers were doing was behaving the way scientists have behaved over history. Scientists observe natural phenomenon and ask questions, other scientists speculate on how to explain the phenomenon, requiring data to be collected and experiments to be conducted. Someone comes up with a solution and explains their results. Other scientists collect more data and perform identical experiments to verify the results. Eventually, there is enough evidence to make the explanation widely acceptable. This new "theory" ultimately leads to new questions and mysteries to be solved.
With the puzzles being solved so quickly, there was a noticeable increase in difficulty with later puzzles, to the point where it was no longer possible to solve these puzzles alone. Later puzzles involved expert knowledge of Shakespeare, T. S. Elliott, HTML coding, CGI scripting, foreign languages (including French, German, Japanese, and Kannada a fairly rare Hindi dialect), art history, religious history, architecture, British cuisine, psychology, Morse code, and the WWII German Enigma code. Someone who knows all of these topics is rare, but in a group of nearly 5000 with Internet access, you can find someone with the expertise that is needed.
Again this parallels the history of Science and Math. There is simply no way for any one person to know everything about a major scientific specialty, let alone be an expert in all disciplines. 19th century mathematician David Hilbert is often considered to be the last person who knew everything about mathematics. Today it is impossible to know everything about mathematics, the topic is too broad. But, it is possible to know everything about one or two specialty topics, and have enough general knowledge to communicate with experts of other specialty topics. The unanswered questions of Math and Science are solved today by groups of experts working together. Einstein consulted dozens of famous scientists and thinkers to formulate his General Theory of Relativity. Today, many new discoveries and inventions are rarely even credited to an individual, but are credited to Universities, Corporations, think tanks, and independent laboratories.
The Reasonable Effectiveness of Mathematics
In a way, this answers the "unreasonable effectiveness of mathematics" problem. Why is it that so much of science can be explained mathematically? Because, so much of mathematics is speculative brainstorming. The unreasonable effectiveness is an illusion.
For every mathematical theory which has gone on to be a good fit for reality, there are hundreds that are just theories and do not fit any reality. Thus we see that of all of the mathematical philosophies, Inventionism (also known as Constructivism), has much more merit when we consider the "hive mentality" of mathematicians. The biggest argument against Inventionism was that the "creation" of mathematics would be culturally biased and would not fit with reality as well as math actually does. When a diverse group with different biases and expertise get together, bias tends to get weeded out. The most elegant solutions eventually get universal acceptance. The invention of the zero is a perfect example. Indian mathematicians discovered that using a place holder made multiplication and division much easier. This innovation spread to neighboring Persia and Babylon, who quickly adapted it. Our "Arabic" numerals came from them. Once a good idea gets invented, it spreads quickly.
Another argument in favor of Inventionism is the fact that there is some cultural bias in mathematics. Despite its universal appeal, we made a mistake adapting a base 10 numbering system. A base 8 or base 12 system would be much more efficient. We are stuck with base 10 because of cultural bias, it is universally accepted because most of us have 10 fingers to count with.
Mathematics and the Brain
I started this inquiry into what mathematics is by comparing two popular theories, the old school "Platonist" view and the new school where math as a product of the brain. The "hive mind" model as a creator of mathematics kind of puts a major dent in both of these theories.
The Platonist view that mathematics is too useful to be invented, gets hit with the realization that when you have thousands of inventors, the good bubbles up to the top and the weak falls off the scope. Under a "hive mind" model, the Platonist argument that Inventionist math would be culturally biased and limited to what we can understand both prove faulty. The good news is that the underlying mysticism of a Platonic realm that we can communicate with and explore no longer needs to be a physical reality, it can remain just a metaphor.
On the other hand, those that say that mathematics is a product of the brain believe that mathematics is ultimately limited by what the brain is capable of understanding. If mathematics is the total of what all mathematicians understand, then the "hive mind" is capable of far more than any individual brain. This leads to a bizarre possibility that there are mathematical objects or concepts which are only understood in the collective unconscious of the mathematical community, but which are too complex for any individual mathematician to comprehend. This possibility is something the Platonists can conceive of and understand.
Could there be a mathematical idea so complex no single mathematician could understand it, but as a group, the world community of mathematicians might be able to understand and work with this idea? None exist yet, but there is at least one possible candidate*: Artificial Intelligence.
The Difficult Problem of making Artificial Intelligence a Reality
Since I have managed to mention "Artificial Intelligence" twice already (in two completely unrelated contexts), I thought I would finish up with a brief introduction to the topic. A more detailed look will be forthcoming in a future essay.
While the idea of creating an artificial intelligence can be traced back through the centuries, the first person to define the idea concretely was Alan Turing, the theoretical inventor of the computer. It seems the idea of making a computer that could "think" is as old as making computers that could only do simple math.
Turing proposed an experiment popularly called the "Turing Test" in which you type messages on a terminal, and either a person or a computer is on the other end. If a computer can imitate a person so flawlessly that it is impossible to tell whether they are real or not, then it is said that the computer program is capable of thought. I will not go into all of the details and philosophical problems associated with the "Turing Test", there are plenty of other great sources.
Suffice it to say that in the fifty plus years since the Turing Test was first proposed, we have not even come close. There are too many problems that have to be overcome first. A major one is natural language processing. Have you ever used a computerized translation program to read web sites in other languages, such as babelfish.com or worldlingo.com? Have you found such programs severely lacking? This is the natural language processing problem. This problem reared its ugly head back in the 1950's when the Defense Department tried using computers to automatically translate intercepted Russian communications. Translation was thought to be an easy problem, enter a Russian word, look up its English equivalent and print the results. The translations were nowhere near as comprehensible as that produced by a human translator. It turns out that translation is far more difficult than just looking up words. For example, read the following paragraph:
If the balloons popped, the sound would not be able to carry since everything would be too far away from the correct floor. A closed window would also prevent the sound from carrying since most buildings tend to be well insulated. Since the whole operation depends on a steady flow of electricity, a break in the middle of the wire would also cause problems of course the fellow could shout, but the human voice is not loud enough to carry that far. An additional problem is that a string could break on the instrument Then there could be no accompaniment to the message it is clear that the best situation would involve less distance. Then there would be fewer potential problems. With face to face contact the least number of things could go wrong. (Bransford and Johnson, 1972)
Do you understand it? If not is there a word that you do not know? There is not a single difficult vocabulary word in the paragraph. So, why are you having trouble with the paragraph? The problem is that you do not recognize the context that the author is talking in. Need some help? Click here to find what this paragraph is talking about.
In order to process natural language, computers need to understand the world around them first. There is the the central problem: how do you make computers understand? Considering the difficulty of programming something so simple we do it without even noticing, gives you just a glimpse of the enormity of the artificial intelligence problem as a whole.
I believe that creating the first artificially intelligent program will require an organized "hive mind" style coordinated effort with many computer programmers, linguists, neurobiologists, and behavioral psychologists working many decades on the project. The first one to create a satisfactory intelligence will be billionaires in no time at all (with such a motivation, it might actually happen within a century).
How Artificial Intelligence can work is something too complex to be solved by a single individual, and if and when it becomes a reality no single individual may be able to understand how it works. It will be a mathematical object too big for an individual brain to comprehend, only understandable by the "hive mind" that created it.
Conclusions
There is an aspect of the Artificial Intelligence problem that has some bearing on our pursuit of the nature of mathematics: Computers have a difficult time understanding simple language, something that comes easy to us. Computers have an easy time solving math problems, something that often times seems difficult for us.
There are two ways to analyze this observation. First, we could conclude that Computers and Humans have such completely differing natures, computers will never be like humans and humans will never be like computers. If so, then mathematics is counter to human nature which explains why it took so long to develop.
The other possible conclusion is that computers are way below us on the intelligence chain. This is the conclusion we must accept if we believe that computers will one day have human like intelligence. If this is the case, then mathematics is too easy for our complicated brains. There is evidence to support both conclusions. Either way we can see that math is not difficult, it is merely contrary to our normal way of thinking.
So far in the quest to answer the question, "What is Mathematics?", we have managed to at least to say what mathematics is not. Mathematics is not mystical, nor can it define itself. It is not a product of individual human minds, but rather a product of consensus among all mathematical thinkers. Ultimately mathematics is something that has been created over time as a means of conceptualizing the natural world. We should not be surprised by its effectiveness at doing what it is designed to do.
What then is mathematics? It is the study of patterns, or more accurately it is a language to describe patterns. Identifying patterns in the world around us comes almost as easy as identifying objects in the world around us. Realizing this is the key to breaking our difficulties with math.
In Part 3, I will look at how the brain does math, and how mathematical thinking evolved. This leads to mathematics as the study of patterns, and why some math is easier than other math.
* A Reader S. W. had this comment: When
you say that AI may be something that is only understood by a 'hive' mind, isn't
that true of many other things? Today there are many technical projects that
require the skill of many different experts (i.e. the human genome). Why is AI
any different? Presumably, someone could have knowledge in all the specialty
areas you list for creating AI. Or maybe humans are not smart enough for that?
Not yet anyway. I think in order for you to be correct about AI, you would have
to acknowledge other 'projects' as being only understood by a hive mind.
I never said AI is the
only example of "Hive Mind" in science, but I completely forgot about
the Genome Project, which fits the bill partially; a single individual can
understand the mechanics of genes and sequences, but no single individual will
ever be able to know how all the genes work and what they do (which is
the next phase of mapping the Genome). One day a giant database will contain all
of this information for us to look up.
Another reader had this example: In your article on "What is Math Part 2" you ask if there is any too complicated to understand, except by the hive mind. I believe the proof of the classification of all finite, simple groups presently consists of over 15,000 pages of journal articles. The problem was worked on for over thirty years, and the chief organizer recently died, so it might be said that no human truly comprehends the entire proof. Work is underway to make it more comprehensible. See "The Classification of the Finite Simple Groups, Number 2" by Daniel Gorenstein and Richard Lyons.
Source:
Bransford, J. D. and Johnson, M. K. "Contextual Prerequisites for
Understanding: Some investigations of comprehension and recall" Journal
of Verbal Learning and Verbal Behavior, 61, 717-726.
Further reading:
kurzweilai.net
is the home page of author Raymond Kurzweil which talks about the present and
future of artificial intelligence and includes Turing Test type experiments.
