Real and imagined mathematics

Do we discover mathematics, or do we create it? Though this question is an old one, it has fascinated me lately. Firstly, an operational definition of mathematics appears necessary to discuss the subject. I would define mathematics (and I believe reasonably so) as the investigation of the properties of logical systems. These logical systems have several attributes: definitions (the meanings of various terms and operations in mathematics e.g. axis, addition), axioms (statements we take as givens e.g. the shortest distance between 2 points is a straight line), and theorems (implications of the axioms). The job of the mathematician is to deduce conclusions (on significant occasions labelled 'theorems') from the definitions and axioms. Seeing as such conclusions follow directly from the axioms and definitions via a logical progression (e.g. axiom: shortest path between two points is a straight line; conclusion: a curve is not the shortest path between 2 points) are we discovering mathematics? It would seem reasonable to say we are discovering the implications of the definitions and axioms, as those inevitably ensue within any system with given definitions and axioms. However, seeing as the axioms and definitions were decided upon by ourselves, one might also argue that we create the definitions and axioms and therefore create mathematics.

The issue with this line of reasoning, it seems to me, is that while we may agree on certain axioms and definitions, I don't believe this implies we've created them. We have simply arrived upon certain ideas and agreed upon them. Before man defined what a line was, did the concept not exist? I think it's important to specify that I am not talking about real world approximations to the concept of an ideal line, which undoubtedly existed well beforehand, but rather the concept of an ideal line itself. Is it fair to say that we arrived upon this concept and accepted it, or that we invented it? Indeed, to say we invented it would seem to imply that the definition of a line depends entirely upon the mental capacity of human beings. This might imply by extension, that if our mental processes functioned differently, the definition of a line would change, a notion that seems utterly at odds to with me. We might call it differently, and refer to it in different ways, but the notion of a line remains the same in my opinion irrespective of human existence.

If we examine the other line of reasoning (which I am more inclined to), that we discover mathematics, this would imply that mathematical concepts have an independent exist of their own thereby allowing us to discover them. The immediate question here, is if mathematical concepts have an independent existence, where do they exist? Here we find ourselves returning to platonic ideas: the theory of forms. The theory of forms states that abstract concepts such as mathematical ones have an independent and immaterial existence outside any parameters we associate with the physical world, such as space and time. This is a hard notion to ever accept, and one for which by definition we can never produce empirical evidence. Nonetheless, it works for me.

Metapost: The End of Junior Year

This final quarter, my blogging has changed in its structure and approach. Rather than dive straight into my opinions on an issue, I have organized my posts to explain the issue objectively at first and provide all the necessary information before then expressing my stance on the subject. This change I believe has been brought about by my work on my Junior Theme, in which I was required to explain an issue pertaining to the USA without expressing any opinions on it at all. The Junior Theme assignement taught me the value of empathy for the reader: assuming the reader does not know the subject and therefore explaining it to them adequately. If I do not inform my reader on the subject, it is useless for me to express my opinion on it at all.
This change in my blogging was important this quarter as I adressed some complicated subjects which required initial explanation. My post "The singularity or not?" adressed the ideas of the futurist Ray Kurzweil. Rather than arguing for or against his ideas immediately, I took time to clarify what they were and the concepts surrounding them before expressing my opinion on them: "Before continuing further, it is important to define the meaning of artificial intelligence. Currently it is split into two kinds: weak A.I., which has already been achieved to an extent, and strong A.I., which has not... ." In this first paragraph of my post, I define the difference between weak and strong A.I. (artifical intelligence), so that the reader knows exactly what it is that I am critiquing. My criticism turns out not to be of A.I. as a whole, but the idea of strong A.I. My prior explanations were important to making my post clear.
In my most recent post, "Down the rabbit hole", I once again make sure to explain the issue I am discussing. The post discusses the constant state of flux scientific ideas are in, with particular reference to the recent experimental evidence against Supersymmetry theory. To clarify my ideas, I make sure to provide an explanation of Supersymmetry and its potential importance as a theory: "Super-symmetry is a theory that relates elementary particles of one spin to to other particles that differ by half a unit of spin and are called 'super-partners'. This symmetry, if proved, would have the potential to help eliminate many known problems and quandaries in modern day physics." This brief explanation of Supersymmetry ensures that the reader understands the relevance of it to my post.
While I believe my empathy for the reader has improved, I have room to strengthen my arguments via more frequent reference to articles and sources. My post "Dreamers", which discusses the 'american dream' and its applicability to rich people, could have benefited from a reference to a dictionary definition of the 'american dream', to support my argument that its defintion prevents it from being available to rich people. Likewise, my earlier posts on the progress of my junior theme could have been made more meaningful to readers with links to sources I was using for the project. Without links to my sources, my posts were less effective in showing the reasons behind my thought processes. Overall, however, I believe my blogging has progressed well since the beginning of the year.

Down the rabbit hole

We've come a long way since Copernicus dared to propose a sun centered solar system. We've moved past Newtonian mechanics. We've arrived at Einstein's theory of relativity. We've probed even further into the workings of quantum particles. And today we find ourselves trying to tie it altogether into one grand and unified theory. Unfortunately, whenever we think we've arrived there, or believe we've come close, we haven't. New data always comes along and shatters our previously "solid" ideas. The standard model today for particle physics is known to be incomplete. It fails to explain phenomena such as gravity and the abundance of matter versus anti-matter. For many years the strongest contending theory proposed to solve such issues was 'super-symmetry'. Super-symmetry is a theory that relates elementary particles of one spin to to other particles that differ by half a unit of spin and are called 'super-partners'. This symmetry, if proved, would have the potential to help eliminate many known problems and quandaries in modern day physics.

However, supersymmetry, which predicted the shape of an electron to be egg like, has benefited from little experimental proof, as recent experiments have measured the electron to be (as far as we can tell) spherical. http://www.bbc.co.uk/news/science-environment-13545453 Seeing as data has appeared which contradicts a prediction of supersymmetry, physicists may find themselves having to rethink the theory. It would seem we are back to the drawing board again. Our worldview and scientific understanding never seems to cease changing. Will it ever? Will we ever understand the universe completely and come to the end of mankind's scientific exploration? I doubt it. The designer, if any, sure made it a challenge.

Dreamers

The term 'the american dream' brings many images to mind. Chiefly, however, it signifies the rise from rags to riches. It is the possibility of this phenomenon which supposedly marks the USA as unique: the opportunity for the poor to rise to wealthy. This is the definition of the 'american dream', and ought not to be confused with the general dreams that americans have.

A dream implies a fantasy about that which one does not have, and therefore to experience a dream, one cannot already be in possession of the object (material or not). To experience the 'american dream', which refers to the dream of obtaining wealth, one cannot already be in possession of wealth as it would then no longer be a dream but a reality. Therefore, those born into wealth in the upper class cannot experience the american dream. They are already rich, and wealth is no longer a dream. They are certainly able to experience and pursue other dreams, but not the dream coined as the 'american dream'.

People's perception of the 'american dream' may vary. Some may view it as the pursuit of wealth, others of happiness and others of success. However, this variation in perceptions does not change the true definition of the term 'american dream'. The term coins one specific dream only, irrespective of perception and its definition is in itself not open to interpretation. The dreams of american society, however, are. Seeing as the american dream has a precise definition, it is therefore open only to those born into sub-wealthy conditions.

The singularity or not?

Several weeks ago in Time magazine, I read an article which gripped me and set my mind working intensely for days upon end. The article focused on the visions of a respected science writer and inventor, Ray Kurzweil, who is closely affiliated with a movement known as the 'singularitarians'. 'Singularitarians' predict the arrival in about thirty years of a technological revolution called the 'singularity' which will change human society as we know it. This revolution will arrive in the form of artificial intelligence. The article is right here http://http//www.time.com/time/health/article/0,8599,2048138-1,00.html.

Before continuing further, it is important to define the meaning of artificial intelligence. Currently it is split into two kinds: weak A.I., which has already been achieved to an extent, and strong A.I., which has not, and many including myself would argue, never will be. Weak A.I. involves the creation of computers and robots which can replicate human behavior: our speech, motion, and the like. Strong A.I. refers to the creation of machines which not only mimic the physical actions of humans, but which are self aware and think just as we do. Ray Kurzweil bases his prediction for the coming of strong A.I. in 2045 off the idea of exponential growth. He examines the rate of progress in computing power over the past hundred years or so, which has grown exponentially (that is, the rate of improvement has become faster itself), and extrapolating off this growing rate of progress, predicts that computers will be made by 2045 which fully replicate the structure of the brain and are therefore conscious just as we are. He believes, in fact, that they will surpass human intelligence.

Such ideas make for fantastic headlines which draw readers magnetically. There is something quite mind churning and provocative about the idea of conscious machines. Singularitarians such as Kurzweil however, are computer scientists and engineers, and coming from such backgrounds ground their quest for A.I. in a fundamental philosophical assumption which they fail to truly examine themselves: the belief in a metaphysically materialistic worldview. By materialism I do not refer to anything relating to consumerism, but to the metaphysical assumption that the foundation of all reality is physical matter, and everything, including the human mind, is a product of physical matter and its interactions. Proponents of strong A.I. believe that the human mind, or 'essence', if you will is wholly reducible to the interaction of the brain's material parts. It can therefore be recreated with robotics.

Many casual readers of the Time article would likely fail to question this assumption, or rather would look away repulsed at the concept of conscious machines, unable however to provide a logically coherent argument against its possibility. My stance against materialism and the possibility of strong A.I. arises from what many philosophers of mind have termed the 'hard problem of consciousness'. The hard problem of consciousness refers to the logical explanatory gap in how one is able to describe all the working parts of a physical structure, such as the human brain, and then explain how, due to its physical interactions, it begins to feel and become self aware. Neuroscience is able to find the brain's nerve correlates of various feelings and thoughts, but is unable to explain how these nerves actually cause conscious experience. If one takes any arrangement of matter, and then proceeds to make it interact in any way, the laws of physics tell us nothing more than the physical interactions which should then ensue. There is nothing in any law of physics which tells us when matter should begin to have conscious experiences. There is no 'Newton's fourth law'. Moreover, any progress in physics will also yield only new laws of physical interactions, which will still leave this explanatory gap: why is it that when physical matter is arranged in a certain way, consciousness, as if from nowhere, arises? The philosopher of mind 'David Chalmers' is perhaps the strongest proponent of this problem. Here is one article of his in which he summarizes the issue. It is long and tough to read, but if one wants to it is well worth the time http://http//consc.net/papers/facing.pdf I think it is important to mention that the hard problem of consciousness is recognized not just by philosophers but by a number of notable quantum physicists today. The esteemed american physicist Henry Stapp sees this problem as the achilles heel of materialism as a metaphysical worldview. He is himself a proponent of mind body dualism: the metaphysical belief that the mind is separate from the body, and in his theory, acts through the brain.

Dualism is an old philosophy formulated by the french philosopher Descartes. He was most famous for his famous words "I think therefore I am", alluding to the idea that since he was able to think he could not doubt that his mind existed. He could however doubt that the physical reality around him existed, as his senses of sight, hearing etc. could all be illusions. He theorized that the mind was an immaterial substance, or soul, which controlled the material body. Dualism fell out of fashion with the development of Newton's mechanics. Following this development, the world was believed to be completely deterministic: all future events were caused directly by past ones, and so could be predicted to infinity. If some immaterial entity were to act upon the physical world, it would violate this fundamental postulate of Newtonian physics. With the coming of the twentieth century, this view was shattered. Indeed all twentieth century physics is so far removed from everyday experience, it is hard to accept if one is not acquainted with the field. Einstein showed, astoundingly, that time is in fact non linear, and travels at different rates for different observers. Subsequently, with the advent of quantum mechanics, the concept of determinism was overthrown. At the quantum mechanical level, events are somewhat random, and cannot be predicted for sure, but only given a probability of happening. That is to say, nothing at the subatomic level of physical reality is for sure set to happen. There are only probabilities. This opens up the door to dualism again. Consciousness, if a separate entity, could perhaps influence this randomness. This is the angle physicists such as Henry Stapp take. He is not the only one, and the noted Oxford University physicist Roger Penrose has devised his own theory in which consciousness is not the product of the brain but a fundamental entity of the universe itself. His ideas are heavily criticized by his materialistic peers, but seeing as materialism is a common prejudice among scientists today brought up with such a world view, and scientists are only beginning to investigate the true nature of consciousness, all bets are off at this point.

Am I a proponent of dualism? Perhaps, I haven't quite decided. I personally find the view of neutral monism more compelling: the idea that matter and mind are not separate, nor the same, but two different properties of some other more fundamental reality (the famous late twentieth century physicist David Bohm and 17th century philosopher Spinoza took a similar stance). One thing I'm sure of though is that I'm not a proponent of materialism, and do not think I'm a robot, as proponents of the quest for strong A.I. would have me believe. The study of consciousness is in its infancy, and I believe singularitarians among others should take a second to question, though not necessarily reject, their materialistic worldview. True scientific understanding cannot be advanced when prejudices go unexamined. I'm sorry this post was so long, but I had a lot of ideas to put out there.

My junior theme and materialism

For my junior theme I have decided to explore the topic of american materialism. However, the initial proposed question, "why is american society so driven by materialistic gain?" was too broad in scope and unclearly defined. I needed to find a specific aspect of america's materialistic culture from which to approach the issue. I chose advertising. The statistics for american advertising expenditures are startling, with the spending figure rising to $131 billion in 2010. A figure such as this begs the question as to the reason for its magnitude. In exploring the reasons for such a figure, my junior theme will focus on america's consumer culture, which drives such high levels of advertisement. Other factors including americans' time spent before televisions and other forms of entertainment will be examined, but all such pastimes really fall under the scope of american materialism as well, allowing my junior theme to explore the topic extensively.

Determining my junior theme

I am currently considering two possible directions for my Junior theme. The first involves an examination of american consumerism and the USA's materialistic values. I will try to answer the following question: why is american society so motivated by materialistic gain? Though the question may need to be defined more specifically, answering this question will require examining, among other things, the history of the USA pre and post the second world war and the changes in philosophies the USA underwent. I will look at the 'american dream': the dream of rising from humble to wealthy that is supposedly the defining dream of american citizens, and examine reasons for its purely materialistic nature.

The other possibility I am considering for my Junior theme entails examining the causes of the USA's current economic crisis. Although the answer is seemingly obvious: too much was borrowed leaving the country in great debt, I will look at the government's tax policies and main areas of spending which drove it to borrow so much. In doing so I would hope to answer why the USA is in its current economic situation. The weakness of this topic as aJunior theme question is that the historical scope is small. At the end of Clinton's administration, the USA's economy had a surplus, and so the reasons for the economic crisis can be traced back only to the beginning of the Bush administration, which was for the most part responsible. One way to overcome this shortcoming for my Junior theme might be to examine the USA's history of debt and borrowing as a whole, and compare it with the actions of George Bush's administration in more recent years.