What does it meant to have sufficient evidence to know something?

While this is not exactly an astronomy question, it is a science question. And, at least in the last few weeks, we have been getting slightly out of hand with some of our discussions, which keep returning to a difference in what we believe/know. So, I thought that it may be good to start a conversation here, where we can keep it streamlined, organized, and civil. This will also keep us from spamming the front page with our discussion.

As a note, I would like to keep this a religion-free argument, since we are trying to have a productive conversation, and do not want to offend anyone. We also do not need this to discuss this subject. So, how to we know things? How do we separate those things from what we want to believe or want to know? And what what do we need to have sufficient evidence ? I think that these are excellent places to start, establish what we agree on, and go from there.

First principles.

I suppose one might start with the question "What is the nature of consciousness?"... Good luck with that. I just make the claim "I think, therefore I am", which, in my view, is a pretty compelling argument, and just leave it that.

Next, start by understanding the universal, logical language of math, so individuals can communicate unequivocally, but that isn't as easy as it sounds. "one plus one equals two"... Really??... PROVE IT. I've seen the proof. It's page after page of set theory and is criticized to this day.

The only practical thing is to assume a generally accepted set of mathematical and logical principles and use those to construct arguments to prove theories, being always mindful of the list of fallacies. If YOU adhere to those principles and you're opposed, fuck 'em. It's not YOUR fault that THEY'RE stupid. If THEY use fallacious argument to "prove" some nonsense you must call them on it. They will generally resist because those using fallacious arguments typically have an agenda, in which case, fuck 'em, It's not YOUR fault that THEY'RE closed-minded.

If you use these first principles to prove an assertion, one can reasonably claim "I know this to be true".

I ascribe to the viewpoint that in scientific matters we only really achieve a high level of confidence. Mathematics permits absolute certainty in formal proofs, but we are fuzzy human beings considering them. Moreover, there are niggly points like the impossibility of proving that Peano's axioms are consistent!

I see science in general as a Bayesian process, where we start with some prior and keep modifying it with evidence. Our beliefs become extremely peaky distributions when we are very sure about a theory, and justifiably so.

I watched a superb lecture yesterday which throws a lot more light on many topics relating to how humans make sense of the world (and how we are trying to make machines mimic our extremely powerful capabilities to do so). It's actually a lot more interesting than the title suggests!

How to Grow a Mind: Statistics, Structure and Abstraction

I am so happy to disagree with 3 of you at the start.   One,  "Keep religion out";  so you have sufficient evidence there is no GOD.  Let's hear it.  Two, "Logical language of math";and "Mathematics permits absolute certanty"; so you assume time and distance are continuous as logical and certain.  Let's hear that too.   You already have my answers, you will find them in the forums right here.   So the question is ; you 3 commented already  ,  do you really want to find THE answers or did you JUST want to comment ? 

Watch that video link and you will be a little closer to understanding how to figure out how the world works!

There is nothing continuous about time and distance. SEE: "Plank Time", "Plank Length".

There is an essential graininess to space-time.

Well, I wanted to keep religion out of it because we don't need religion to discuss all of the relevant topics here, and because if we start trying to make absolute claims about religion, then the conversations will degrade into only a discussion of that, which doesn't help us get closer.

And I agree that mathematics is the language of logic. But in order to show something is false, you need one of two cases. You either imagine a situation where all of the premises are true but the conclusion is false, or you assume what you are trying to disprove, and then show that that assumption leads to a logical contradiction. Here is an example:

You gave the example of an infinite number of non-zero derivatives being impossible. So, I postulate you consider the example f(x)=e^x. In this situation your premise is true, but conclusion false, because this is a real function with infinite non-zero derivatives. So, you need to add another premise, or change the theorem.

There are many ways of knowing. The various sciences have different thresholds of statistical certainty before they will accept something as a fact.

In physics, that threshold is generally 5 sigma (standard deviations). That means that there is less than a 1 in 3.5 million chance of being wrong.

I subscribe to Jaynes' interpretation of how we infer truth about the world. This is that what we do is justified as some sort of approximation to Bayesian reasoning, where we start with a prior (the selection of which is a deep philosophical question, with some useful principles to guide it), and then modify that prior with Bayes rule using each subsequent observation to reach posterior probabilities which quantify belief.

I am not actually sure if particle physicists explicitly use this or (horror of horror) use frequentist statistics, but often an uninformative prior means a "significance level" (a badly motivated concept) is associated with an actual posterior probability which is a meaningful quantity.

An important principle (which is often conveniently ignored) is that it is unwise to preclude any possibility with a prior. Better to give it a low probability. Interestingly, quantum mechanics suggests that the truth about the world is very similar: there is nothing too unlikely to happen, but many things have very low probability of happening.

There is an interesting discussion of whether particle physicists could improve their use of statistics here (turns out my suspicion above was largely true!)

https://errorstatistics.com/2012/07/11/is-particle-physics-bad-science/

Hi dnivickers; I just reread your note.  It is so easy to missunderstand.  I did not say "infinite derivatives are impossible" .  What I tried to say was that continuous time and distance requires infinite derivatives and therefore continuous time and distance are impossible.  Your example is good, so is a sine function.  But it turns out every funcion requires infinite derivatives, even a stright line if these two are continuous.  

Well, Paul, I am just curious about why the need for infinite derivatives implies  that continuous time and distance is impossible. Because we have established that we are fine with having functions with infinite derivatives. What what about the infinite derivative is so bad? I think this is the point that we are getting caught up on. Because I have never, personally, found that assuming something had infinite derivatives lead to a logical contradiction.

If you could guide me to that contradiction, I would appreciate it.

Mathematics has the luxury of dealing with absolute precision, by creating abstract models which model infinite objects (apparently, and probably) entirely adequately. The real numbers are a set of ways to divide the rational numbers into left and right. As such they are continuous in a sense that can be described in a mathematical theorem ("the real numbers with the topology generated from the usual order is a connected topological space", for example).

In physics we use the idea of real numbers as a convenient general tool: we NEVER have access to a real number, merely quantities with some uncertainty (eg 1.6021766208 x 10-19 C +/- 0.0000000098 x 10-19 C, the charge on the electron. We can never know a precise value). The same is true for any distance, momentum, or whatever.

But whether or not you believe space is infinitely divisible (physicists don't really - they all realise that large scale physics cannot be right at very small scales), the ideas of continuity at all scales which we have access to is very useful.

For example, if an object flies across a room, you will not go wrong by assuming it takes a continuous path rather than jumping at any point. If it is a series of little steps (for which there is no evidence), the steps are so small as to be indetectible.

This contrasts dramatically with, say, the momentum levels in a confined system. These are quantised, and the distinction from having continuous values is detectible and important.

"Mathematics has the luxury of dealing with absolute precision, by creating abstract models which model infinite objects (apparently, and probably) entirely adequately." Look at that statement. The math is GREAT; it just does not apply to our universe. That is what "sufficient evidence" means !!!   One must start out correct, that means the first statement.   TRY IT ONCE !!!

So where is the experiment where maths doesn't work?

Count your fingers; that count is discrete.  Count the hairs on your head, that experiment give discrete counts.  Write any number. that experiment only gives discrete numbers.  WHERE is your experiment that states the answers are different than I just stated ?  EVERY THEORY has values were the theory is wrong.  And even that number of wrong theories is discrete.   

All numbers are defined in terms of integers. First the rationals as pairs of integers, then the reals as ways to divide the rationals. Integers appear a lot in physics. Where real numbers appear they are always uncertain, so have a rather different character to definite. The mathematics of real numbers (and other objects built on them) remains relevant.

I keep trying to drag you back to science. Where is the experiment that shows real numbers are NOT a suitable basis for a concept in physics where they are used? Merely guessing space is discrete does not prove it is.