I've had the idea for a long time that the field of psychology would be much better off if it was mathematically formalized the way the physical sciences are. As of now it is essentially qualitative, which leaves room for all kinds of subjective bias. Beyond that theories often lack falsifiability and predictive capability, which is to say that they are not really scientific. Psychology needs its Keplers, its Galileos, its Newtons to come along and get the ball rolling with a strong mathematical model that can be built on. That would essentially raise the whole field above pseudo-science to actual science.
Original Idea:
I'm not sure how to explain this idea. I shall simply describe it as it developed in my head.
I was thinking about vector spaces and quantum mechanics. With vector spaces the idea was to use them as psychological models by associating each measurable psychological quality with an axis in the basis. Then for any given person we could measure each of these psychological qualities and represent them as vectors in the space. Math on the vectors (either basic operations or functions) then allows us to find relationships between these qualities and other qualities, or to look at larger effects caused by groups of them together (essentially the sum of several vectors).
With quantum mechanics I noticed that a quantitative psychological model should really yield probability distributions for what is likely to be occurring in a given person, the same way the square of the wavefunction describes the probability of measuring a certain value of a physical observable. Hence I wondered if the mathematics behind quantum mechanics could similarly be adapted to a model of human psychology.
For those that don't know what I'm talking about with vector spaces/QM:
Problems:
The thing about vector spaces is that they're only one type of algebraic structure that you could use for such a thing, and not necessarily a good choice. I essentially picked them at random because I wasn't very familiar with other structures when I first had this idea. Why a vector space? Why not a ring, monoid, group, field, etc? Why even limit it to a known structure? Algebraic structures consist of a set of elements and some number of operators defined on those elements satisfying given conditions. Before we can answer what type of algebraic structure might work best to develop a quantitative psychological model we have to answer:
1) What are the elements we are interested in (e.g. the psychological attributes/qualities/etc.)?
2) What are meaningful operations on these elements?
The simplest would be to simply use numbers as elements, for example probability distributions (i.e. numbers) as mentioned above - this basically reduces the algebra to normal arithmetic. We don't *have* to do this though, for example Plutchik's wheel of emotions can be modeled as a set of elements with a binary operation on them (I suppose also as a mapping between a Cartesian product and another set, but who cares). The point is that this level of abstraction allows us to not even necessarily need to measure things numerically and to deal with psychological concepts themselves.
The other issue is with the quantum mechanics approach. The "mathematics of quantum mechanics" is essentially an equation that relates the probability distribution (more specifically the wavefunction) and the energy levels (more specifically the eigenvalues) of what we are measuring. First issue, while there is a psychological equivalent in probability distrubutions, there is not in energy levels, or at least I cannot think of one. Secondly, we still have to find a meaningful equation to relate these things, so in a sense we're back to square one as far as trying to find quantitative relationships between psychological variables.
Development:
In any case, I've come to the following general idea of how quantitative psychological modeling might work:
1) Determine what it is we wish to predict from what measurable quantities.
3) Find a meaningful mapping between the elements being measured and the elements being predicted, using the operations defined. (i.e. find equations relating important variables)
4) Profit.
Conclusion:
Ok, so I realize I haven't actually found anything close to a useful quantitative psychological model yet. Sorry. What I think I have done is broken this problem down to the most abstract level. Obviously *simple* equations do not exist to relate things like probabilities of given thoughts or behaviors, which is why I think mathematical formalization of the subject might see more progress if people stepped outside of the idea of statistics and real numbers.
The grandiose idea is that such a model, once developed, could be used in a clinical setting to make predictions reliably, rather than based on subjective bias, as the system is fully scientific: testable, falsifiable, and predictive. This would most likely take a lot of information gathering to begin to be able to make accurate predictions, and hence may not be particularly useful unless every relevant aspect of someone's life was monitored (intrusive). That's kind of irrelevant though: the Navier-Stokes equations may be too hard to solve for many real systems, it doesn't make them useless overall (this seems like a good analogy). A quantitative, fully scientific psychological model would be a big step forward for the whole field, IMO.
Anyhow, this is the current state of my thoughts on it. There is a lot more I could say on it, for example about MBTI and vector spaces (or other structures). I'll save that for another post (edit: no I won't, it's at the bottom). Does anyone else have any ideas though? Things I'm particularly curious about:
1) What are the fundamental elements/qualities of psychological analysis (i.e. the elements for my mathematical system)?
2) Is this an (in)appropriate way to develop such a mathematical formalization?
3) What other sorts of approaches or mathematical tools might be considered?
4) Is there some fundamental reason that this will not work? I'm sure some would argue that behavior and thought transcend quantitative analysis, however I'd rather not just take that as an axiom. I'm not into mysticism.
For all the people who undoubtedly know more about math/modeling/etc. than I do, please do tear me down if I'm talking out of my ass about any of this or am misunderstanding anything.
* After I wrote all this it occurred to me that I should have considered Bayesian networks as well. I don't have time for that now, and I haven't really thought about them in this context. Just throwing the idea out there.
Ok fuck it, the thing alluded to above about MBTI and vector spaces:
MBTI essentially reduces all the measured variables (what you answer on the questionnaire) to four axes: I vs. E, N vs. S, T vs. F, P vs. J. This then groups people into sixteen simple categories. This is probably good for people, but in other terms it's just diluting the actual data.
Consider instead that each question on our MBTI questionnaire is it's own axis in a orthogonal basis spanning the vector space. Each subset of the questions then spans another subspace of that vector space. So for example, all the questions dealing with T vs. F-ness create a T vs. F subspace. The system can be degenerated to MBTI simply by summing all the vectors in each of the four subspaces, yielding four new orthogonal vectors spanning another subspace. Obviously your overall MBTI type is given by a vector in the space, the components of which are your scores on any given question.
This approach seems superior for several reasons: first it doesn't simplify people into sixteen discrete types, but views them as a fluid spectrum of positions in an n-dimensional space. Second, it allows for other mathematical and psychological relationships to be found. In a fluid spectrum it's meaningless to just look at S/N or T/F traits and such, you can look at any particular vectors in the spectrum you want. Every subspace is a different axis in itself to be examined. Thirdly, there may be more complicated mathematical relationships to be found there, like meaningful functions on the vectors or other interpretations due to the mathematical representation of a traditionally non-mathematical thing.
As I said above though, vectors don't seem special in any particular way, they're just the most familiar idea that occurred to me. Something else would most likely work better.
Obligatory xkcd:
Original Idea:
I'm not sure how to explain this idea. I shall simply describe it as it developed in my head.
I was thinking about vector spaces and quantum mechanics. With vector spaces the idea was to use them as psychological models by associating each measurable psychological quality with an axis in the basis. Then for any given person we could measure each of these psychological qualities and represent them as vectors in the space. Math on the vectors (either basic operations or functions) then allows us to find relationships between these qualities and other qualities, or to look at larger effects caused by groups of them together (essentially the sum of several vectors).
With quantum mechanics I noticed that a quantitative psychological model should really yield probability distributions for what is likely to be occurring in a given person, the same way the square of the wavefunction describes the probability of measuring a certain value of a physical observable. Hence I wondered if the mathematics behind quantum mechanics could similarly be adapted to a model of human psychology.
For those that don't know what I'm talking about with vector spaces/QM:
-Vector spaces: Just imagine any kind of 2-d or 3-d coordinate system. For purposes of discussion these are vector spaces. Now imagine in a 3-d coordinate system using each axis to measure a different psychological quality. Now imagine doing this but with a coordinate system of any number of dimensions (thousands, millions, etc.).
-Quantum mechanics: When you do the same experiment many times you don't necessarily get the same result, even under the same conditions. The likelihood of measuring a given result is given a distribution of probabilities among all the possible results. Similarly, people placed in a given situation will all react or think differently, however some reactions/thoughts are more likely than others. It also forms a probability distribution.
-Quantum mechanics: When you do the same experiment many times you don't necessarily get the same result, even under the same conditions. The likelihood of measuring a given result is given a distribution of probabilities among all the possible results. Similarly, people placed in a given situation will all react or think differently, however some reactions/thoughts are more likely than others. It also forms a probability distribution.
Problems:
The thing about vector spaces is that they're only one type of algebraic structure that you could use for such a thing, and not necessarily a good choice. I essentially picked them at random because I wasn't very familiar with other structures when I first had this idea. Why a vector space? Why not a ring, monoid, group, field, etc? Why even limit it to a known structure? Algebraic structures consist of a set of elements and some number of operators defined on those elements satisfying given conditions. Before we can answer what type of algebraic structure might work best to develop a quantitative psychological model we have to answer:
1) What are the elements we are interested in (e.g. the psychological attributes/qualities/etc.)?
2) What are meaningful operations on these elements?
The simplest would be to simply use numbers as elements, for example probability distributions (i.e. numbers) as mentioned above - this basically reduces the algebra to normal arithmetic. We don't *have* to do this though, for example Plutchik's wheel of emotions can be modeled as a set of elements with a binary operation on them (I suppose also as a mapping between a Cartesian product and another set, but who cares). The point is that this level of abstraction allows us to not even necessarily need to measure things numerically and to deal with psychological concepts themselves.
The other issue is with the quantum mechanics approach. The "mathematics of quantum mechanics" is essentially an equation that relates the probability distribution (more specifically the wavefunction) and the energy levels (more specifically the eigenvalues) of what we are measuring. First issue, while there is a psychological equivalent in probability distrubutions, there is not in energy levels, or at least I cannot think of one. Secondly, we still have to find a meaningful equation to relate these things, so in a sense we're back to square one as far as trying to find quantitative relationships between psychological variables.
Development:
In any case, I've come to the following general idea of how quantitative psychological modeling might work:
1) Determine what it is we wish to predict from what measurable quantities.
Some abstract examples:
a) Thoughts from behavior.
b) Future behavior from past behavior.
c) Future thoughts from past thoughts.
d) Behavior from thoughts.
2) Define meaningful operations on these elements.a) Thoughts from behavior.
b) Future behavior from past behavior.
c) Future thoughts from past thoughts.
d) Behavior from thoughts.
3) Find a meaningful mapping between the elements being measured and the elements being predicted, using the operations defined. (i.e. find equations relating important variables)
4) Profit.
Conclusion:
Ok, so I realize I haven't actually found anything close to a useful quantitative psychological model yet. Sorry. What I think I have done is broken this problem down to the most abstract level. Obviously *simple* equations do not exist to relate things like probabilities of given thoughts or behaviors, which is why I think mathematical formalization of the subject might see more progress if people stepped outside of the idea of statistics and real numbers.
The grandiose idea is that such a model, once developed, could be used in a clinical setting to make predictions reliably, rather than based on subjective bias, as the system is fully scientific: testable, falsifiable, and predictive. This would most likely take a lot of information gathering to begin to be able to make accurate predictions, and hence may not be particularly useful unless every relevant aspect of someone's life was monitored (intrusive). That's kind of irrelevant though: the Navier-Stokes equations may be too hard to solve for many real systems, it doesn't make them useless overall (this seems like a good analogy). A quantitative, fully scientific psychological model would be a big step forward for the whole field, IMO.
Anyhow, this is the current state of my thoughts on it. There is a lot more I could say on it, for example about MBTI and vector spaces (or other structures). I'll save that for another post (edit: no I won't, it's at the bottom). Does anyone else have any ideas though? Things I'm particularly curious about:
1) What are the fundamental elements/qualities of psychological analysis (i.e. the elements for my mathematical system)?
2) Is this an (in)appropriate way to develop such a mathematical formalization?
3) What other sorts of approaches or mathematical tools might be considered?
4) Is there some fundamental reason that this will not work? I'm sure some would argue that behavior and thought transcend quantitative analysis, however I'd rather not just take that as an axiom. I'm not into mysticism.
For all the people who undoubtedly know more about math/modeling/etc. than I do, please do tear me down if I'm talking out of my ass about any of this or am misunderstanding anything.
* After I wrote all this it occurred to me that I should have considered Bayesian networks as well. I don't have time for that now, and I haven't really thought about them in this context. Just throwing the idea out there.
------------------------------------------
MBTI and Vector Spaces:
Ok fuck it, the thing alluded to above about MBTI and vector spaces:
MBTI essentially reduces all the measured variables (what you answer on the questionnaire) to four axes: I vs. E, N vs. S, T vs. F, P vs. J. This then groups people into sixteen simple categories. This is probably good for people, but in other terms it's just diluting the actual data.
Consider instead that each question on our MBTI questionnaire is it's own axis in a orthogonal basis spanning the vector space. Each subset of the questions then spans another subspace of that vector space. So for example, all the questions dealing with T vs. F-ness create a T vs. F subspace. The system can be degenerated to MBTI simply by summing all the vectors in each of the four subspaces, yielding four new orthogonal vectors spanning another subspace. Obviously your overall MBTI type is given by a vector in the space, the components of which are your scores on any given question.
This approach seems superior for several reasons: first it doesn't simplify people into sixteen discrete types, but views them as a fluid spectrum of positions in an n-dimensional space. Second, it allows for other mathematical and psychological relationships to be found. In a fluid spectrum it's meaningless to just look at S/N or T/F traits and such, you can look at any particular vectors in the spectrum you want. Every subspace is a different axis in itself to be examined. Thirdly, there may be more complicated mathematical relationships to be found there, like meaningful functions on the vectors or other interpretations due to the mathematical representation of a traditionally non-mathematical thing.
As I said above though, vectors don't seem special in any particular way, they're just the most familiar idea that occurred to me. Something else would most likely work better.
Obligatory xkcd:
