Is doing Calculus based on irrationality ?

[Langbein]

Is doing Calculus based on irrationality ?

I have always used to think at Mathematics and Calculus as something based on pure logic and that is a logical deductive system.

Then I tried to sit down and think a little bit over how Calculus is done and I got a bit surprised.

If I do some Algebra, or if I solve some equitations I’m usually able to do so using some rules or some patterns that will usually lead to the solution (If it can be found.)

If I should make a program that should solve any kind of mathematical problems (that I were able to write) the way it would work would to apply a set of rules in some “logical deductive pattern”.

On the other hand, if I am solving some composite integrals, that will be difficult for me, but that might be easy for others, I observe myself working in a quite different way from just applying rules, the rules is applied, but not in a straight forward way.

When you take a look at that “integral thing” there will normally be no straight forward way of solving it.

First you go trough a set of basic integration rules to se if any of these basic logical structures will work. If this will not work you try some other logical structures like integration by parts, integration by substitution etc, etc.

But what is this search of that logical structure that will fit the best for the certain task, does this search itself follow a straight forward logical pattern ?

Can it be doubts that solving difficult integrals will involve some creativity ?

What is then “creativity” ?

Will the process of applying creativity on the problem involve some other tasks or methods than just applying straight forward logical deductive rules ?

After applying some creativity, the next phase will typically be some “after rationalization” to check if the last logical model that were found will fit in, in a logical deductive patern.

But what is this process of searching for the applicable logical model ..is the process itself managed using logical rules ?

 

[honestrosewater]

I'm not sure where the philosophical part is. If you want to know how humans solve such problems, you can ask in https://www.physicsforums.com/forumdisplay.php?f=149" . There are some cog-sci/comp-sci people there, methinks. If this is the case, a mentor can move the thread; you shouldn't start a new one.

Was there a different question, e.g., what other objects does creativity depend on?

Also, it sounds like you just said how to solve your problem: you go through some list of rules that you know. What is creative about that (whatever creativity is)? If conditional processing counts as creativity, I think you'd have a hard time finding a program running today that isn't creative (or couldn't be made creative with the equivalent of a few keystrokes).

In fact, since the basic idea is simple, and I don't want to do the work I'm supposed to be doing (), we can start writing one of these "creative" programs now. I'll make up a mini-programming language that I think will be clear to English speakers.

To start, you have two variables that will be input to the program: the problem and the list of problem-solving method names. Name these "the problem" and "the method list", respectively. Call a member of the method list a "method name". You don't have to worry about defining them now. You find out what they are when you run the program.

You also have to define a function that will check how well a given method will work for a given problem. I'm not sure how you expect that to work, and I never took calculus, so I'll just assume that it works for now and not care how it works. Say that it takes as input parameters a problem and a method name, in that order, and returns, for now, simply a "yes" or "no" to indicate whether the method will work for the problem. Let's name it "check" and run it by saying "check" followed by by its parameters in order, and let's name the string returned by check "the response".

The last thing we need is a function that actually applies a winning method to the problem. We'll assume that its inner instructions are taken care of already too and just name it "apply" and have it work the same way and have the same parameters as check and return a string that we'll name "the solution".

For each method name in the method list, check the 
problem and the method name. If the response is yes,
then apply the problem and the method name, write 
down the solution, and sit back and relax. If the 
response is no, then carry on with the iteration.

Well, I assumed that you know how to handle a "for each" iteration and conditional, write, and other common things, but those can be broken down further too. I actually used a while-loop in disguise, but meh -- you're a human, you understand what "sit back and relax" means. You would just rephrase the above to say "while there is no solution, ..." rather than "for each...", and set things to exit the loop when you get a solution. Anywho...

Do you see how this sometimes tedious programming stuff works?

 

[tacman]

In short, yes, doing Calculus involves a combination of both logic and creativity. While there are certain rules and patterns that can be applied in solving mathematical problems, there are also instances where creativity and problem-solving skills are necessary to find the most efficient solution.

The process of solving complex integrals, for example, requires a combination of both logical thinking and creative problem-solving. While there are basic integration rules that can be applied, these may not always lead to a solution. In these cases, one must think outside the box and try different approaches until a suitable solution is found.

Creativity in mathematics can be defined as the ability to think outside the box and come up with innovative solutions to problems. This involves using different methods and approaches that may not follow a straight forward logical pattern. It is this creative thinking that allows mathematicians to solve complex problems and come up with new theories and formulas.

However, it is important to note that even in the process of creativity, there is still a level of logic involved. Creativity in mathematics is not a random process, but rather a systematic and structured one. The search for an applicable logical model may involve trial and error, but it is still guided by logical thinking.

In conclusion, while Calculus is based on logic, it also involves elements of creativity and problem-solving. The process of solving complex problems may involve a combination of different methods and approaches, but ultimately, it is the logical application of these methods that leads to a solution.