The 'method' of theoretical research

[saltine]

In an experimental scientific research process, you could state the hypothesis and the methods you would use to conduct the experiment. The experiment gives you data. Then you use statisitics to check whether the data disproves or is unable to disprove the hypothesis. However, in a theoretical research (could be in mathematics, or philosophy), what is the 'method' when there is no notion of experiment? Say you have a hunch that a mathematical relation could be proven, but you don't know exactly how, then what is the 'method' of the research process? Do mathematicians in this kind of research only publish after they have the results? (Unlike scientists who could talk about the setup of the experiment, the number of test subjects needed before they get any data from the experiment.)

 

[dx]

Theoretical research is essentially a creative process, like art. Just like there is no particular method to produce a good painting (although there are standard tools), there is no method of doing original theoretical work.

 

[mordechai9]

I am happy to see this question because I find it very interesting myself. It is somewhat astounding that the mathematical engineer/scientist can reach a new, observable result purely on the basis of mathematics. We can call this the "mathematical method", a kind of specialized form of the scientific method, or perhaps an entirely different method altogether.

First of all, you say that there are no experiments in the mathematical method. I don't entirely agree. For example, say you plug some numbers into a formula, providing some concrete relations between observable (or notional) quantities. This is tantamount to doing an experiment, because if you had done an actual experiment, you would wind up (in the end) with a set of numbers or relations between quantities -- say, between temperature and pressue. So plugging numbers into an equation seems a bit like doing an experiment.

Furthermore, you can program the equations of scientific processes into computers and then run simulations which show the results of the theory under certain circumstances. Again, this is exactly tantamount to performing an experiment, only in this case, the concept of "plugging in a few numbers" is massively extended. In fact, with a bit more programming, you can produce graphical results on the computer screen which may duplicate very realistically the kind of visual results you would see in a laboratory. So I do not agree that mathematics is entirely devoid of experimentation -- however, the experiments it allows are not founded on the laws of reality; instead, they are founded on the specific, perhaps peculiar mathematical basis which the experimenter controls from the beginning.

Anyways, the mathematical method really provides results through the logic extension. Starting with realistic postulates and representative deductions, an equation may be established, say, for the vibration of a series of connected masses. Then these equations may be operated on LOGICALLY -- i.e. preserving the truth and integrity of the original representation -- to provide a previously "unseen" observation. The key point, in my mind, is that the mathematical research method relies on the logic structure to start with some representations -- verify the truth of these representations -- and then extend or manipulate these representations mathematically to provide previously unseen information. I could go on a lot longer but I feel this is too extensive already.

 

[saltine]

I think I am now seeing the parallel between theoretical inquiry and the experimental process. It seems that often times the bulk of theoretical research is in the phrase of coming up with the question, and the "experiemental" process is trivial, while in scientific research, the importance of the inquiry is shadowed by the experiment process.

In this perspective, a scientific process is not any easier. Knowing what question to ask, identifying the data to collect and designing experiement are all done in the theoretical inquiry stage, where there could be equally no 'set method' for doing so.


An imaginary example of how mathematical research is like doing science:

Initial question:
The researcher: I suspect that a certain property of a system can be derived mathematically.

The refined question that is equivalent to a scientific hypothesis:
Hypothesis: this relation can be solved analytically by direct substitution.

The equivalence of an experiement:
Researcher does the math.

The equivalence of results:
Results: Yes it works; or no, direct substituion is not enough.

Perhaps one main difference is that in theoretical research, since the experiement stage is much faster, there would be much more "experiements" done per topic. Say the research discovers that direct substition does not work, so he turns around and trying a different approach. He could be doing the second try on the same sheet of scratch paper. If he had documented it formally, that would have been the second "experiement".

 

[mordechai9]

I really like this portion of your reply:

An imaginary example of how mathematical research is like doing science:

Initial question:
The researcher: I suspect that a certain property of a system can be derived mathematically.

The refined question that is equivalent to a scientific hypothesis:
Hypothesis: this relation can be solved analytically by direct substitution.

The equivalence of an experiement:
Researcher does the math.

The equivalence of results:
Results: Yes it works; or no, direct substituion is not enough.

That seems to be realistic. Also, you are right on target with your idea of the "initial question". Basically, even using the mathematical techniques, the research process still replies on the researcher's intuition. The researcher suspects something can be gotten from somewhere, somehow, and then they work towards it.

However, some of your other ideas could be improved. For example, the experimental research process can often be devastatingly difficult. There are so many unexpected issues that arise when trying to make precise experiments of complicated things. Often times, these experiments are completely intractable. It requires a huge amount of ingenuity to figure out how to measure and design an experiment for certain things. There are indeed trivial experiments, just like there is trivial mathematics. But that is not representative of the whole process.

Also, the theoretical process can be hugely more involved then you may believe. I have read that the initial investigations into string theory (see Lee Smolins "The Trouble with Physics") required humungous amounts of mathematics to establish the initial results -- equations which continued over perhaps hundreds of pages. This is just an inkling of the difficulty that may be involved.

 

[fahad03092]

Could you please anyone help me to give me details idea and steps about theoretical research with example.


I am trying to find out a relationship in a process. In literature there is no existing relationship between this process parameters. So I collect data from published journal and plot them together and plot a two dimensional graphs to prepare a equations. Is my work included in theoretical research? What I can do in my next step? Can I publish this equation? Please help me.

 

[saltine]

Hello fahad03092,

I think researches can be classified in several ways, for a particular research, it could have characteristics of more than one term:
o Scientific
o Theoretical
o Survey
o Applied

A theoretical research has a characteristic where the object being studied either does not exist or cannot be physically observed or measured. However the steps are equivalent to scientific research, in that:

1) You still need to come up with some sort of hypothesis or explanation
2) You still need to test whether that contradicts anything that you know

I think that in your case, if your hypothesis is that the process parameters are related, once you have collected the right data and showed that they are related, the research is technically complete. However, in reality it would beg the immediate question:

"Why are they related? Does one cause the other? Are they both caused by a third parameter? Or are they the same just the ratio people have used traditionally?"

Once that question is answered, another question would come:

"How would someone use this knowledge to improve the process? What is the impact of this knowledge?"


I think if you could follow through all the questions, it should be ready to be published, but sometimes relations could be significant enough that they are worth publishing without explanation or application. I think this depends on what the journal wants to include, you could tell by checking the papers in the journal and see how far those researches got when they were published.

 

[fahad03092]

Thanks a lot for your powerful and scientific answer.


I actually collect data from different journal who are works on Plasma spray process and used materials calcium phosphate. As the author change there is a change of equipment. Change of equipment has a little effect of process parameters. If I go for same equipment then I have only 5-10 data. That's why I ignore the effect of equipment change to collect data as much as possible. After that I plot data and find a trendline which is acceptable in terms of physics for this process parameters. Though in some graphs the fitted equations have r square less than 70%. Do you think should I need to strict on r square 70% or over?


I am able to explain the graphs in terms of physics or from the supporting literature. But I just scare about the acceptance from the reviewer? Do you have any idea or any feedback to make it more accepted?


Thanks a lot in advance.

 

[saltine]

Hi, that is beyond my domain knowledge. I am interested in what others have to say though.

 

[chiro]

Hey saltine.

Research can be really haphazard. It's something that can be really disordered and erratic and you pick up new things that you never would have expected from a variety of sources: it's definitely something that is not strictly linear.

With mathematics though, you will probably have an idea of what to delve deeper into. The good thing about mathematics is that once you know the language, you can pretty much write what you want to find out in that language. With the paragraph you wrote down, you'll find that some author (or set of authors) out there has dedicated a lot of their time in "writing novels" dedicated to that kind of paragraph.

For example you might find that the math you wrote down is an optimization problem, and then you find that there are dozens of books and journal articles on optimization which take you to the next level and you continue this kind of thing.

The moment you write some kind of definition down, or conjecture (in the language of mathematics), you are creating a constraint. The constraint itself might be broad, but it is still a constraint none the less.

Once you have that constraint, you can use that as you have written it, or you can apply even more constraints and look into researching the more constrained problem. If you find something interesting that pertains to the more highly constrained problem, chances are it may help you with answering the broader problem.

If you are doing this kind of research (mathematical and/or theoretical in a mathematical nature), don't put all of your attention in a narrow field. If you have a problem and a field has been created that answers your problem, then that's great! (You're a lucky researcher!). If the problem is tough to answer, then don't fret: many problems are like that.

If it is like that then read widely. Chances are you'll get new insights from different fields that might help you. I'm not recommending you become an expert in every field, just that you at least know the basic ideas of what understanding a field is trying to accomplish: that will give you more to think about and help clarify further direction for where your research should take you.

 

[chiro]

Also, this is probably an unorthodox suggestion, but sometimes other fields are ripe for giving you ideas and suggestions that might help you solve your problem. You'd be surprised how many examples and analogies out there in a wide variety of areas can help solve what at first might seem a seemingly unrelated problem.

Nature can have a way of providing solutions or insight into things that can help solve problems that you might otherwise not think about. Nature has a habit of being a large part of inspiration for creating new areas of science in general.

 

[twofish-quant]

In an experimental scientific research process, you could state the hypothesis and the methods you would use to conduct the experiment. The experiment gives you data. Then you use statisitics to check whether the data disproves or is unable to disprove the hypothesis.

That's a textbook description, but in fact it's much messier than that. One problem with that description is that to interpret any sort of data, you have to have a mental framework which influences the interpretation of that data

The other problem is that most textbooks descriptions of science philosophy miss the fact that things are rarely "prove/disprove." For example, if I had experimental results that seemed to show that the world was flat, my first reaction would be that you messed up the experiment. Now if I have a dozen experiments that indicate that the world is flat, then I'd start changing my mind, but the reason I think that the world isn't flat is because of the hundreds of thousands of experiments that people have done before.

However, in a theoretical research (could be in mathematics, or philosophy), what is the 'method' when there is no notion of experiment?

Mathematics is very different from physics. Theoretical physicists are model makers. Someone gives me a bunch of data, and I come up with a mathematical description of that data with some ideas about what they should look for next.

Also a lot of theoretical physics has something similar to experiment. You put together a model, you run it on a computer, and you see what pops out.

 

[twofish-quant]

The researcher: I suspect that a certain property of a system can be derived mathematically.

The way that it works in physics theory, is that you ask the math people if they have any math tools by which a certain property can be derived mathematically. If they say "we don't know" then ask yourself "so what?" Suppose this is mathematically true, what are the consequences? Suppose this is not mathematically true, what are the consequences?

If it turns out that it doesn't matter, then you write a paper saying that "this doesn't matter". If it turns out that it does matter, then you write a paper saying that "this does matter, and could the mathematicians look more closely at this problem."

 

[twofish-quant]

There's some really interesting questions about "who decides what is science?" Suppose you come up with a description of a "scientific method" but then I go up and say "but that's not what we actually do in the university." So what happens then? There is a philosophy that says that science is whatever scientists do, but then you have to ask who is a scientist.

Something that I find interesting is that a lot of descriptions about what scientists do is rather recent. The idea that science is about disproving ideas is a philosophy called logical positivism. The interesting thing is that logical positivism was invented in the 1920's. Thomas Kuhn's work on science was written in the 1960's.

 

[twofish-quant]

There's some really interesting questions about "who decides what is science?" Suppose you come up with a description of a "scientific method" but then I go up and say "but that's not what we actually do in the university." So what happens then? There is a philosophy that says that science is whatever scientists do, but then you have to ask who is a scientist.

Something that I find interesting is that a lot of descriptions about what scientists do is rather recent. The idea that science is about disproving ideas is a philosophy called logical positivism. The interesting thing is that logical positivism was invented in the 1920's. Thomas Kuhn's work on science was written in the 1960's.

 

[Bourbaki1123]

The interesting thing is that logical positivism was invented in the 1920's. Thomas Kuhn's work on science was written in the 1960's.

And ET Jaynes' work was even more recent.

 

[Bourbaki1123]

There's some really interesting questions about "who decides what is science?" Suppose you come up with a description of a "scientific method" but then I go up and say "but that's not what we actually do in the university." So what happens then? There is a philosophy that says that science is whatever scientists do, but then you have to ask who is a scientist.

Something that I find interesting is that a lot of descriptions about what scientists do is rather recent. The idea that science is about disproving ideas is a philosophy called logical positivism. The interesting thing is that logical positivism was invented in the 1920's. Thomas Kuhn's work on science was written in the 1960's.

A lot of the fields of experimental design and inferential statistics is relatively recent as well.

 

[920118]

There's some really interesting questions about "who decides what is science?" Suppose you come up with a description of a "scientific method" but then I go up and say "but that's not what we actually do in the university." So what happens then? There is a philosophy that says that science is whatever scientists do, but then you have to ask who is a scientist.

Something that I find interesting is that a lot of descriptions about what scientists do is rather recent. The idea that science is about disproving ideas is a philosophy called logical positivism. The interesting thing is that logical positivism was invented in the 1920's. Thomas Kuhn's work on science was written in the 1960's.

The idea that good science consists in falsifying theories isn't really what's called positivism. The idea that falsification is what scientists should do was developed by Karl Popper as a reaction against Comtes' positivism. Popper's theory is also completely normative, while Kuhn's is descriptive. What Popper does is basically to assert that induction isn't a valid method for gaining knowledge. He then goes on saying that scientists should discard every theory that has a false implication, while completely ignoring that the falsification of theories in practice has to rely on induction aswell, since a large number of theories have to be accepted as true in order to allow us to point at a specific theory's falsehood. (Sure, the argument is deductive in form, but most of the premises will have to be based on induction anyway)

What Kuhn does is to present an idea of how science actually works, and has been working throughout history. Popper and Kuhn were probably the best known rivals in the scientific rationalist/anti-rationalist debate, where rationalists said that scientific advancement actually is based on good epistemic grounds, and that it moves "forwards", while the anti-rationalists said that shifts in theories can't be completely rational.

Sorry if my english is undecipherable.

 

[twofish-quant]

The idea that good science consists in falsifying theories isn't really what's called positivism. The idea that falsification is what scientists should do was developed by Karl Popper as a reaction against Comtes' positivism

Cool.

One thing that sort of weird is that scientists in fact spend almost no time thinking about the philosophy of science. You just do it, and you leave insights into what you are doing to the philosophers.

 

[Bourbaki1123]

Cool.

One thing that sort of weird is that scientists in fact spend almost no time thinking about the philosophy of science. You just do it, and you leave insights into what you are doing to the philosophers.

It's no good, because then you get all of this muddled philosophizing and then a real scientist comes in and explodes all of the bad ideas! I would say that the best philosophers were also scientists or had a deep appreciation for science. Think of C.S. Peirce or Judea Pearl (with his work on causality) or Daniel Dennett or Frank Ramsey.

Of course, once in a while do you get real technical results arguably inspired by philosophical prodding; I think that the cognitive sciences have a good number of examples, and I would go so far as to say that there is something very philosophical in the production of new hypotheses, especially in more interdisciplinary and young fields. You are forced to conjecture and draw out implications. Of course, the scientist goes a step further and actually tests the merit of the ideas, whereas the philosopher is tempted (and often gives into her/his temptation) to tailor his ideas to evade falsification. Even worse, the ideas are often totally removed from reality.