How can One Generalize a Result?

Question

Suppose we are trying to generalize some result, for example, a well known theorem or some theory. The generalization must improve the result in some way. What are the best tactics to do this?

Let me try to explain it better: In the common way of learning, we start to learn things that are more easy to understand and then we go to more complicated things, for example, in a course of analysis we learn the Inverse Function Theorem (IFT) for finite-dimensional Banach spaces. If we continue to study (for instance if you are an analyst), in some time we will learn about (or at least hear about) the IFT for general Banach spaces. As we can see, this theorem includes the finite-dimensional case.

To do this kind of generalization, we have to try and find some properties that are common to both cases. Moreover, the generalization, in some cases, must contain some key properties that we aren't able to see when dealing only with the standard case. So, how to find these properties? How to know the best way to generalize some kind of property?

If my question is nonsense please disregard it, and sorry about the English.

Answer 1

Some result as you say, is a theorem $t_m$ involving instances of some structure $m$. You'll find a more general result exactly then when you consider a more general structre $M$ and construct a theorem $T_M$ for which $T_m=t_m$.

E.g. you might consider as $m$ the polinomial $x^4$ and you find a theorem $t_m$ given by

$$\frac{\text d}{\text dx}x^4=4x^3.$$

The more general expression $M$ given by $x^n$ with $n\in\mathbb{N}$ fullfills a $T_M$ given by

$$\frac{\text d}{\text dx}x^n=nx^{n-1}.$$

As the last result $T_M$ is valid for all $n$, a less general theorem $T_m$ for $m$ is implied as well. Of course, I have constructed this example in a way in which $T_m$ exactly yields the previous $t_s$.

Now how did I go from $m$ to $M$? Here I abstracted the instance $4$ by a more general object for which I decided the "taking power of x"-process would be sensible too. I understand $4$ to be of type $\mathbb{N}$. For formal proof of any of the theorems $t,T$, I will need some syntax and proves of deduction.

I think the answer to your question "how to come up with the generalisation (of the structure)?" might just be "out of need for a specific problem of interest". And the answer to "how to actually find the structure which generalizes another" might be that one will have to abstract the syntactic rules governing the math you dealt earlies so it fits your new problem.

I tried to sneak in some ideas of model theory in here, but I'm really a physicist and the first example coming to mind when reading your question what the quest of generalizing the theory of electrodynamics to fit in nuclear force phenomena. In a question over many many decades, people eventually reformulated the former to be a gauge field theory with $U(1)$ symmetry group and its generalization to bigger groups $SU(n)$, Yang-Mills theory, now governs the Standard model of particle physics. The insights to consider fairly abstract group structures and their matrix representations was motivated by the observation of the structural similarities found in the particle zoo of the $70's$ and then people tried and tried. I guess I must formulate it as: They ended up with a theory which didn't not work. My conclusion is that there is no general abstraction algorithm, which and generates new theories (which yields valuable results) in finite time :)

Answer 2

I will try to explain using a example:

• At the start of studies, continuous functions could be introduced. For example only for $\mathbb{R}\to\mathbb{R}$, using the $\varepsilon-\delta-$definition.
• Lateron, it is generalized to any $\mathbb{R}^n$ by using norms and then to normed spaces. You see that the absolute value $|x-x_0|$ can be replaced by $\|x-x_0\|$ and gain generality.
• As a last step, you observe that the definition "f is continuous if the fiber of every open set U is open" is just as good. This leads to the definition of topologies.

As a last observation, we see that if you remove axioms from topologies, you lose the ability to define a the idea of continuosity.

This is the basic train of thought for generalization: You start with something that is close to our understanding and try to remove axioms/ideas/requirements etc while maintaining the general idea.

To exactly answer your question, generalizing a result would be the process of eliminating preconditions without changing the result.

Answer 3

If you wanted to know what 50 Fahrenheit is in Celsius, what you're teacher first taught you to do was to subtract 32, divide by 5, and then multiply by 9 in that order. After you knew the answer, the next obvious question is, well how do I find 60 Fahrenheit in Celsius? You quickly begin to realize the pattern, and we recognize that pattern as being a formula: $$C = (F - 32)*\frac{9}{5}$$ That formula can be applied for any F. We later learn that through a series of formula operations, you can create a formula for doing the inverse, Celsius to Fahrenheit.

We wouldn't have been able to do this had we not created the mathematics to support the simple fundamentals of algebra. Likewise, you wouldn't be able to find the area under a curve without integrals and so it goes on. The problem is that you reach a certain point in which you're no longer dealing with formulas but concepts. That has proven difficult to simply contextualize and consequently it makes it near impossible to apply generically. It's my personal opinion that it is not because you cannot, but because our limitations of understanding means we have not yet invented the mathematics to do such things. It would be like Newton attempting to calculate the orbit of a planet without first inventing Calculus. It can be done, but it becomes so monumentally difficult to conceptualize that you almost literally have to first invent the mathematics before it becomes realistic.

I think we simply lack the mathematics to generalize a theorem, but that may one day change.