Financial Mathematics Text

Sunday, December 30, 2007

Our Relationship to Nature from Classical to Quantum Physics

...we have to remember that what we observe is not nature in itself but nature exposed to our method of questioning. Our scientific work in physics consists in asking questions about nature in the language that we possess and trying to get an answer from experiment by the means that are at our disposal.
-Werner Hesienberg. Physics and Philosophy.

Here, I don't think, from this brief distinction it is clear that there's a difference between classical and quantum physics and our view of what scientific inquiry amounts to. The emphasis that is later noted is that "we ourselves are both players and spectators." It is this that seems where quantum physics view of the relationship between the observer and the observed is different from the classical picture. Nature has a set of potentialities which are only actualized when an observation is made. I would like to sketch a metaphorical description of this relationship.

In the view of classical physics, nature has a set of predetermined answers. When we ask nature a question, it searches her database of predetermined answers and finds one that is appropriate. In some cases, none exist so we are forced to ask a different question. The crucial aspect is that the answers are already set in place and it's just a matter of nature finding the answer appropriate to the question.

In the view of quantum physics, nature doesn't have a set of predetermined answers. Rather, when we ask nature a question it is as if it first reflects upon the question and then answers the question. It's almost as if we are asking questions which nature had not considered before and it's compelled to consider the question and answer.

Thursday, December 27, 2007

The Myth of Purpose

It is often supposed that, at least some, human activity is a product of "reflection" and that actions are motivated by having purpose or seeing some end involved. I would like to suggest (as an exploratory hypothesis) that this is not the case and give a rough sketch on how it could have arisen that we think in these terms.

Growing up we are trained into a linguistic practice and part of the practice that we have come to possess includes the question "why". I recall from my days of learning Spanish that the question "why" translates literally "for what". We might think of it as "for what reason" did you do such and such an action. We are seeking an explanation of an event in terms of what ends one pursued which in some way caused the actions in question.

Part of being trained in a linguistic practice is learning to give appropriate responses. It's not just that I am asked "why" but I'm required and expected to give an appropriate answer, even if there was no reason. This last part is important. It can very easily be the case that there was no "conscious reflection" on my part when I was acting. I may have had no reasons in mind at all when I acted. But since I have been trained in linguistic practice and since "no reason" is often not accepted as an appropriate response, I create a reason. I give an interpretation, after the fact, of what my reasons for acting were even if there was no actual reasons there to begin with.

This possibility seems to raise the question whether or not there are such things as "reasons" at all, that are not mere ad hoc creations after the fact. Part of this may be to observe the etymological roots of such words relating to this issue ("why", "reason", "purpose", etc) but also to explore societies who do not possess "purpose language".

Tuesday, November 20, 2007

On the Variety of Experience

In a previous blog (see here) I put forth a "refutation of empiricism". The problem with this refutation (and I think I was somewhat aware of it at the time but couldn't fully figure out how to vocalize it) is that it fails to make some distinctions between different kinds of experience in favor of how experience has been treated, historically, in epistemology. My overall critique, I think, is accurate in the sense that it points out the failure of epistemology to address important issues due to very specific (and quite honestly bad) simplifications which fail to deal with matters in a more intelligible matter.

Part of my own explorations in the philosophy of mathematics is to accurately characterize how knowledge is obtained via mathematics. Mathematics, whether correctly or incorrectly, has historically been viewed as the prime example of knowledge for which all other sciences can only hope to attain. This is, in my view, still true today. I suggest that it is no coincidence that there is a correlation between how "hard" a science is and how much math is involved, with physics being considered supreme amongst the sciences. The general thesis that "knowledge comes from experience" and "by experience we mean sensations or observations" is clearly incompatible with the view that mathematics is the prime example of knowledge since mathematics fails to have those essential properties. But I would actually suggest that the way in which the empirical sciences actually work is contrary to the way empiricism is often worked out in the philosophical literature. I've noticed that even many scientists fall into the mischaracterizations, as given by the philosophical literature of the very work they do. In any event, these are, more or less, underdeveloped thoughts on my part. What I intend to do is list three kinds of experience. I will not say that these are exhaustive but I do think they better clarify this matter.

1) Experience as "sense impression"

I borrow the phrase, roughly, from Hume. I could very well use "sense data" or "observation" or "perception" (although this word is certainly not univocal; it's possible to "misperceive", for example.) This is the sort of empiricism which I most certainly rejected.

2) Experience as experiment

This more resembles that of what the empirical sciences use. Experimentation was advocated by a number of philosophers including Descartes and Bacon as well as many scientists such as Galileo and Newton (and to some extent I abhor the distinction between these two: scientist and philosopher. I'm not sure why it is we don't count Galileo and Newton amongst the philosophers; I hope this mistake will be rectified.) Here we set up a controlled experiment and record the results of that experiment (which are often measurements of some sort.) It should be clear that my critique of empiricism does not address this form of experience although I still have some considerable doubts as to the claim "all knowledge comes from experience", even in this sense of the word. I do think that some knowledge comes from experience of this sort.

3) Experience as practice or "know-how"

This sort of experience is a practical sort of experience. It's typically an activity of some sort which is practiced and "with experience" one comes to "know how" to perform this activity. Playing an instrument, riding a bike, performing surgery: these are all activities which require time and practice in order to learn and become better at. They require "experience". Again, my critique of empiricism does not address this form of experience but I would conjecture that the form of knowledge obtained via this kind of experience is a different kind of knowledge. The general concern of epistemology is propositional knowledge, for example: I know that 1+1=2. In this case we have something different which I'm not entirely sure how to classify. I think it may be, perhaps, reducible to knowledge by "acquaintance" or at least related to acquaintance in some way. What this seems to be are examples of the form: I know how to ride a bike.

As mentioned before, it's not entirely clear that the above is an exhaustive list but it should clarify my rejection of empiricism more by clarifying what sense of "experience" I am rejecting.

Monday, November 19, 2007

Dr. House and Abductive Reasoning

Every Tuesday night I got to watch my favorite TV show (and by "favorite TV show" I mean "the only show I really bother to watch"), House. If you haven't seen this show please do so as it's great. Hugh Laurie is absolutely brilliant as an actor in the show. A brief rundown of the show is that it's about a diagnostics department that is headed up by Dr. Gregory House (played by Hugh Laurie). He works with a team of doctors whose job is to diagnose difficult cases that other doctors are having difficulty with. House is a renown doctor but his antics are a little less desirable (and all the more amusing!)

A patient will come in with a set of symptoms and House and his team will begin making suggestions as to what the patient has. What House and his team are using is abductive reasoning. Abductive reasoning looks for an explanation or hypothesis that, if true, would (best) account for a set of facts, in this case symptoms. Often two diagnoses are appropriately given although House seems to have a preference for simpler explanations. Often several hypotheses are suggested none of which are particularly liked by House. Further tests are run at this point in order to rule out (or potentially confirm) some of the hypotheses. In some cases, all hypotheses offered by the group are ruled out, so House and his team search for new explanations.

Sunday, November 18, 2007

The Relationship Between Excluded Middle and Eternalism

Committing oneself to the principle of excluded middle (universally) suggests that statements of the following sort must either be true:

"Tomorrow it will rain or not rain."
"I will die in the year 2021 or I will not die in the year 2021."
"The Mets will win the 2009 World Series or the Mets will not win the 2009 World Series."

My ability to make these claims rests on the principle of excluded middle which states for any proposition P either P is true or ¬P is true. It seems as though to commit to the excluded middle is to commit to a view on the future which suggests it's decided in some sense. This view is commonly referred to as "eternalism" which views the future just as fixed as the past. It's not a difficult view to comprehend. If we think of time as being analogous to a spatial dimension then it's no different than viewing different points of space as being fixed and decided. Of course we can always ask whether or not the past and present are really as decided as "common sense" might tell us but that would only be a greater cause for concern. It appears that one who is committed to the universal applicability of excluded middle would either have to contend that statements about the future are not propositions so excluded middle does not apply or to commit to an eternalist view with respect to time.