Mathematics is an analytical tool that can be used to find a solution to a question. However, the appropriate mathematical tool to be applied is a function of many variables among which might be subjective rather than objective. Hence one could approach the relationship a specific mathematical tool or formulate one that will best solve the problem.

Mathematics is not an analytic tool but a description of how the universe works using mathematical language. You can describe, in perfect mathematical detail, how a tree falls in the forest but the tree falling in the forest does not need mathematics to fall.

mathematics is a language which help us describe the world around us. We use the signs of mathemtics to create our story of the world and events outside us. using the same signs can help us deliver our understanding of how things work and what affect other factors. it is easy using math. to tell your story and it is much understandable by others who use the same language.

@John

@Eliyahu

Description of the known phenomenon in equations is one thing, searching understanding beyond the known, and pushing the frontier of knowledge is a different one.

Actually i am preparing a course on Scientific Methodology in Finances. And i got suck, as i believe i have to teach students how to select the proper tools for their research purposes, that should be both efficient and effective. Asking others how they do-it sounded like a good idea. Since finance students training in mathematics is not as extensive as for mathematics students, it's obvious they may face situations where they have to study mathematics further. As mathematics are a huge field, finding and understanding the optimal mathematical approach in a determined time-frame may be tricky.

What will be the main area/topics your student will conduct a research? What kind of research they will make? Is it by analysing data-set by time or finding connections or correlations between factors? I think you need to determine first the type of research the drow out of it the specific math. Tools required

Dear Eliyahu,

Your question is a mighty good question, an answer would ease my effort a lot. I cannot say, my students will work in banks, insurance, public finance, companies of all sorts, and it is very hard to predict what research will they do. Even those few that may go further towards PhD i cannot predict what types of research will they do. So, i cannot "predetermine" the methods, i have to help them find the best ones for their future purposes.

I am looking for some procedure on finding an adequate mathematical approach on a variety of mainly practical problems. I do not want to limit them, i want them to learn to expand their knowledge and keep up with the new.

Best would be to build up some Mnemonic they would remember easy, like "Objective, Data, Other Researchers Approach" ODORA. or something like a checklist.

So, i am investigating now as general idea, how other researchers select their tools.

Best regards,

Bradut

First i must say i find this discusion exciting. To my opinion and upon my experiance i think students should now how to generate a target function with all its restriction. we should train them the way of thinking about a "life problem" as a target, like opening a big closed box in trying to understand what is inside? how it works and such questions.

certainly, they must check other researchers work and way of thinking, and make the necessary changes to suit their problem. i find your menemonic ODORA quit good and easy to remember. i would have add to it the F letter at the end for Formulate to express the "life problem" with math. tools as target function with restrictions.

Where would you place the Formulate in the ODORA? FODORA, OFDORA, ODORAF?

i would place it at the end because it shuold be at the end of the proccess. Still after formulating we need to solve (make calculations, generate derviativs, run some regresions linear or non linear, use the SOLVER of the Excel, drow some graphs and so on) the problem in order to get some insights out of the formulation stage. But maybe it doesn't "sounds" so good as a menmonic. thus maybe we need to think of a better word....

I like to simplify as much as possible, I will search some words that may fit in a better mnemonic.

Interesting topic

Hi, I am a banking risk manager with 12 years experience currently moving from the business to science. Your question is something I was struggling with quite often throughout those years.

My thoughts on this are following:

1) When studying financial mathematics, statistics, econometrics I should focus on "philosophy" behind approaches and on examples of usage. It would be good if - for example within the framework of courses in statistics - universities provided information on a wide spectrum of methods with examples of applications. There is great variety of problems encountered in "real life" - so it would be good to hear about a lot of methods, without necessarily mentioning details.

Secondly, if one understands the "philosophy" behind a method, then it is more likely that - even "creating own niche" is possible.

2) Within my career I have seen many times analysts in the area of retail credit risk, who tried to reinvent extrapolation of vintage delinquency curves. Last year, talking with students in insurance I found out that such methods are described in scholarly papers on insurance mathematics under the name of "chain ladder analysis".

This example shows that "OR" of your ODORA is quite important. But you have to know how to ask the question. It is easier if students have background on wide spectrum of methods.

Dear Blazej Kochanski,

The spectrum of methods provided by economic training is more or less limited to mathematics already extensively used in economics. I want students to go beyond those and dare into "uncharted" mathematics whenever the usual mathematics they may know do not provide an adequate answer.

best regards,

Bradut

I would say that getting to know "philosophy" behind the method would not only help to use the method properly, but also would help to invent new or alter existing method if needed. And from practice I would say that it is needed quite often.

Thank you for starting a really interesting topic!

I agree with Blazej. What I'm trying to teach my students is that every problem has 3 components (in order of importance).

a. Asking the right question (seems self-evident, but its not always the case),

b. Knowing where to look for the proper tools aka "the philosophy" as Blazej puts it. It gives you several degrees of freedom within your own knowledge spectrum and makes easier finding someone who knows better, in choosing the right one.

c. Actually using the tools.

I think that its so much easier to train for the use of the tools than to educate for the proper question articulation and the "feeling" of the right tools. Thats why Blazej is right. The grasp of the "philosophy" comes thru experience stemming from the variety of problems faced.

Very interesting topic indeed !

Could we divide the "question" phase and the "looking for proper tools" into some kind of algorithm?

I suspect available data are somewhat a pointer to method (discrete, continuous, fuzzy).

For 99.9%of users I would suggest putting the "question" in google or google scholar.

Find a paper with high citations that uses mathematics. Look at some recent ones that cite it.You will have a good idea of the maths required.

Dear David,

I agree, finding a article that solved the problem already is a really good solution. And most likely someone somewhere has already asked the same question and found an answer.

I cannot help being frustrated about your answer. I definitively agree that available papers on the topic should be studied, but something bothers me about that. Something is missing in this logic... If all questions are answered, why bother?

Best regards,

Bradut

When studying financial mathematics, statistics, econometrics I should focus on "philosophy" behind approaches and on examples of usage.

Knowing where to look for the proper tools aka "the philosophy" as Blazej puts it. It gives you several degrees of freedom within your own knowledge spectrum and makes easier finding someone who knows better, in choosing the right one.

I want students to go beyond those and dare into "uncharted" mathematics whenever the usual mathematics they may know do not provide an adequate answer.

I agree, finding a article that solved the problem already is a really good solution. And most likely someone somewhere has already asked the same question and found an answer.

When trying tosolv a problem in any discipline, the mathematical tool that one might use is a function of the goals that one to achieve. For exqmple, if one is trying to test somehypothesis in social science subjects, the statistical tool one will use will depend on one's objectives whereas in other disciplines the mathematical/statistical tools will be different. However, ones goal is the major factor of which mathematical tool to be used in solving a problem. This question is very interesting!

Let's follow a bit David Peel's idea.

Searching already published work is undoubtedly one step. If the results are intended to be published, the relationship with already published works have to be clearly stated. That implies a deep understanding of researches in the same field and close to it.

Personally i like to keep my work as original as can be, the downside being that citations are unlikely and heavy promotion of the ideas are required. In a sense, in order to achieve impact, research should not stray much from the mainstream, thus the required originality comes from gradual improvements on existing research.

So, David is actually right for most people, especially before having a certain fame, publishing history, and network of related scientists, promoting original ideas is hard and very frustrating, thus existing work clears the path.

For unscientific focused work, mostly business purposes research, originality may be more adequate, as following the trend means being always one step behind the herd, thus being vulnerable. Even if documentation is just as required, a clear relationship with other research is less important compared to adequacy to the purpose. As time in business has a higher value, results being demanded on a tighter time-frame, consuming time with extensive documentation is less efficient. In that environment having a working model based on available data is more useful than having a better model that does not work because data is unavailable timely.

This is why some mathematics are less used, as continuous data is less frequent compared to discrete data. So, in practical economics discrete data mathematics are more likely to be usable. In engineering and other sciences continuous mathematics are usually more adequate.

So i would normally say to my students to avoid continuous functions and all mathematics related to them, as long as they have a discrete approach. If they use continuous functions in economics they should be aware of the build-in approximation, as between two cents there is nothing... unless we approximate. And two and a half products rarely means anything except a product and a unfinished product.

Accounting data uses almost always discrete positive numbers, and it is the basis for most data available at any level. Unless there is no other option this should be taken into account.

Internet searching has been a vast improvement on tedious searching in limited library resources of my time. Nevertheless, it is not a substitute of original thinking and question forming. I usually suggest this workflow:

a. Search for the existence of a similar question with one in hand.

b. If found, is the approach and answer given satisfactory for the purpose we want them? (no point reinventing the wheel time and time again).

c. If yes, use the results as intended. If no, is there a way to approach it differently to reach the result we anticipate? Fresh look of a known question is always welcome

d. If no such question has been asked, what other work in the same domain has been done, and what tools were used? Get a clue of a path already taken. If satisfied pursue it, if not, begin from scratch clearing the path for others to come .

I like to think that ends define the means here. Usually forming a question, an expected answer is formed too. The existence or not of that answer is what we try to establish.

Its true, as Bradut puts it, in business, things are quite different. There the first response is a quick and dirty answer, usually defining the order of magnitude. If a more detailed result is necessary, then more work has to be done, more elaborate reasoning used and more sophisticated tools employed, in much smaller timeframe. But still the answer has to serve one purpose, a decision to be made. So, all the work done has to produce a result having the characteristics necessary to the decision to be taken. i.e. In most cases there is not much difference in percentages like 99,99% and 99,999%. But if the problem in hand has to do with the safety of passengers of a busy airport, counted in millions, then none of them is acceptable.

And, finally, there is a lot of space between 2 cents, financially speaking. Look at the exchange rate quotations for an example.

Interesting topic and one I am not sure I am qualified to answer from a technical perspective but I have thoughts from a philosophical one. For me, the starting point is the purpose of research. My research is primarily macroeconomic, if it can even be called research. The purpose is to predict the decisions of the FOMC on rates, bond purchases etc., to forecast the direction of the U.S. economy, assess the impact of fiscal policy changes, provide insight into the direction of markets etc. The choices I make given time constraints will be entirely different from the choice made by an academic researcher. Let me give an example. One of the key issues for the U.S. and monetary policy is how much of the current unemployment is cyclical versus structural. One way of looking at this problem is the Beveridge Curve linking the unemployment rate to job openings. At the present time, the number of job openings is relatively high given the unemployment rate. I am not a specialist in labor economics and am not going to do some grand exercise in assumed micro behavior but I do note that the minimum wage is now $7.25 an hour versus $5.15 in 2007, which is a 41% increase over six years (but the increases took place from June 2007 to June 2009). The U.S. adopted unprecedentedly long unemployment benefits (up to 99 weeks) versus the usual six months. Both of these developments are likely to lengthen search times or interfer with matching (don't get an entry level job because of higher minimum wage and then not able to gain skills for a better job). The median duration of unemployment rose sharply from around 8 weeks in 2007 to 22 weeks in 2010, which is consistent with longer searches (in 73-5 or 81-2 the median duration of unmployment was much lower for comparable unemployment rates). So far any math I have used is no more complicated than a couple of charts. I think that charting data is massively underrated by many economists, who might leap into more complex statistical methods. Now my next step might be to run some regressions--probably linear to start with--paying attention to nonsense regression indicators (DW or fancier tests). I am not sure where advanced math comes in from my approach. However, this topic approached by a labor market economist might have microfoundations of search costs, investigation of panel data etc. My feeling is the profession spends too much time on formal modeling and math and not enough on looking at data and trying to apply some simple economic principles to tell a story. I also rate history more highly these days than I did 30 years ago!

The number of assumptions we make in an analysis is staggering. Those assumptions are the weak-point of all approaches. We may not even identify all assumptions we made that are wrong.

I was reminded recently that there are two types of investors but this could apply equally to economists. There are those who don't know and those that don't know that they don't know. If we try and fall in the first class and remember that economics is not a science but a way of thinking about problems that may yield some insights, we will be all better off as a subject.

It depends on the field and the strategy that somebody wants to follow each time.

First of all a deeper mathematical background is always needed. You can learn something extra, but if you have not understood the hierarchy of maths you will get lost in non useful equations and other objects.

Then you have to find the mathematical classification for your problem: is it a problem of partial differential equations (PDE)? Is it a problem of identifying a hidden math-property from available data?

Some disciplines are working with two or three mathematical categories, so it is easy to find the tools.

Derivatives theory is just a diffusion theory for variance, ie a PDE section.

Classical Economics everything turns around the Sun of Neoclassical Models for competitive equilibrium, so you need Optimization theory.

It is not so difficult to find the proper mathematics.

The difficult problem is to ask the new and hard questions and to insist on your questions until you manage to answer them.