Food for thought Sets logic and applied science

[galoisjr]

I was just reading an article the other day about the debate in public schools about teaching evolution as an absolute truth as it has been taght for the past umteen years. Not saying that I'm a proponent of creationism or even that I'm not, but there are some serious flaws in teaching it as the absolute truth considering much of it deals in pure speculation...

So anyway, this got me wondering what exactly constitutes scientific proof? Is there even such thing as absolute proof in science? Or are we doomed to only see evidence and not the entire picture?

The biggest problem lies in an area very familiar to us mathematicians: in set theory. Let's say we are given a theory. Now this theory is supported by a possibly infinite set of evidence. However, each single piece of evidence may or may not (more likely than not in my opinion) belong to the set of evidence supporting another theory also. In many, possibly infinite, circumstances, the single element of evidence may belong to a possibly infinite amount of sets each supporting a different theory. Also, given a theory with a possibly infinite supporting data set, there is also is more likely than not a possibly infinite amount of intersections with the data sets of other theories and each intersection may contain many, or, you guessed it, possibly infinite elements...

You can see how this gets old really quick...

Anyway not having looked at as much set theory as I would have liked to lately, I came to a question that I could not answer. And I really do not know if it is answerable...

Suppose you are given two sets whose intersection contains infinitely many elements. Does this restriction require one set to be a subset of the other? (if they are both subsets of each other then we have equality and thus the trivial conclusion) And also if not, can you think of an example?

For instance, an example of the affirmative would be that the integers and rationals have an infinite intersection and the integers are indeed a subset of the rationals.

 

[galoisjr]

Ok nvm this answer was obvious... I over complicated it.

 

[galoisjr]

Oh and btw the answer is obviously no and can easily be seen by drawing your basic coordinate axes and arbitrarily choosing two intersecting sets in which neither is a subset of the other... Which seems to imply that a scientific theory can never be proven no matter how much data there is to support it... What a bummer.

 

[chiro]

To me truth is something that everybody can agree on. Absolute truth is something that absolutely everybody can agree on.

To me it's all relative and too many people make it about certainty when it's not since having incomplete information means you are working under uncertainty and that you will be making inductive inferences based on data that you don't have.

It might help you to understand how statistics is advancing to answer your question since it with its foundations in probability take a look at this very thing with sets corresponding to events.

You might be interested that in statistics a lot of people "accept" hypotheses when they should "fail to reject them based on the evidence".

The distinction is subtle but it's important because when people hear "accept" they think of the statement being true which is not the case: we always have two kinds of errors in hypothesis testing namely Type I and Type II errors that deal with the situations where we get evidence from our data to suggest one thing but actually the real attribute is the opposite, and this is something that a lot of people who use statistics fail to remember.

The best we can do is to allow anyone that cares enough to do the experiment themselves and have as much transparency in the data, the process and the setup.

Science is not meant to be a way to obtain absolute truth: it's meant to be a way of systematic decentralized knowledge discovery that is structured in the way of not only doing something specific, but also for describing, documenting and communicated the context of that thing.

Absolute truth means that you know all the information in some capacity or another and this is just plain impossible: so the best we can do is work under uncertainty and make the best effort we can to understanding both uncertainty and the relativity involved in dealing with not only incomplete information, but the relativity of different results where the context of each investigation has both mutually exclusive and overlapping details which corresponds to your discussion using sets.

The key though is to have a balanced mindset: have enough confidence so that you can draw the line and stand behind a decision but not be too arrogant to just only hold on to some subset being right and its complement being wrong.

This is actually extremely hard because it requires the same mindset even when in someones limited experience, they are being subjected to evidence through their own experiences and even through other people that they consider credible.

No one wants to be told that they are wrong and this creates even weirder dynamics where some people will even go to lengths to at the least make assumptions that they initially think to be true or even unfortunately as far as modifying the data intentionally to get the output they "expect" to get.

It's hard for people to accept that they are wrong and it's also hard because everybody wants to feel like they are in control and being told that you are wrong translates into a loss of control.

When people feel like they are not in control down to a certain point, protective mechanisms come in and this creates a weird set of dynamics again in the context of science.

There are no doubt a lot of really great scientists that do a lot of great work and have really strong scientific integrity, but this human instinct to feel in control does some crazy things to both the actions and the minds of human beings.

 

[micromass]

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