[Slider thankfully has pointed out some flaws in parts of my argumentation and I recommend you to also read Slider's comment.]

Everybody who wants to have agency over one's live should believe two things. Firstly, there is an objective truth about the world in which we live. Secondly, we can partially discover this world: There are ways in which we can get an approximate understanding of what's true.

However, questions remain even if these things are taken for granted. My which methods can we get to this truth, and how precise can our approximation get?

It's quite clear for me (and I did not think of these issues myself but believe that they are well-covered in classical philosophy) that any method that uses all available information in the most effective and sound way should have logic as one of its tools (without the advantages of logical inference, the method is not effective; if it does not avoid contradictions, it is not sound). Furthermore, assuming that we don't think we already know the truth (which seems not plausible), we should look out for hints from the environment, evidence, which help us to get better insights into the real world.

There are some more things which basically all human beings who are able to survive assume: that we can partially trust our senses, except when they contradict each other or we have other reasons to doubt. These are basically axioms; just as the mathematician has to assume some axioms before beginning to work, nothing about the real world can be derived without axioms (but as soon as we have those axioms, together with our sensory experiences, we can infer much more conclusions about the real world using our brains). Having nothing but logic or mathematics, there are still many possible ways the world might interact with us. So we need these axioms, or if you wish, priors to trust our sensations.

However, choosing those axioms can be tricky. To get information about the whole of our planetary society, our senses are not enough. We need to extend their range using communication technology and media: phones, newspapers, the internet. To be able to talk about what happens on the other side of the earth, we have to extend our trust in our personal sensations to trust in some newspapers or that telecommunication cables faithfully transport information between different places. 

So as soon as people started to globalize their lives, and therefore increase the size of their world relative to their immediate environment, the had to add another prior to their repertoire: What people (or media) tell us about distant happenings should be considered true if there is no reason to think otherwise.

Of course "no reason to think otherwise" is not well-defined, but there are some hints. If the reporting of media about supposed facts is contradictory between different journalists, bloggers, and media enterprises, one should lower one's confidence in the overall system and find out which sources are the reliable ones. If the messages from different sources are not contradictory, this is usually a good sign, except when there's a climate of fear or conformity among these sources.

Today, we seem to be in a difficult situation. There are lots of contradictory voices, at least in the internet, and it is very hard to identify reliable information. It seems impossible to DESTROY some position with FACTS and LOGIC because for every claim, the internet provides people that have objections that can't be dismissed out of hand. At the same time, established media outlets lose trust because they sometimes consider pieces of evidence as soldiers and not as gifts and seem to fall prey to the same group dynamics most human beings do when living in a polarized society. On the other hand, there are lots of people everywhere in the political spectrum that are happy to uses these problems to erode trust and deconstruct these institutions in and of itself, until we can't trust anyone anymore.

Maybe it's all become too complex? What if our extended sensations, namely internet and media, are so indirect and have so many middlemen and the topics they try to report on are so complicated that what remains on our end has a low signal-to-noise ratio? What if the resolution of our extended sensations is too low to reliably see the information that lets our beliefs converge against the truths we need so urgently to solve the most pressing problems in a globalized world?

I really, really don't know. I just feel so empty of orientation at the moment, though full of information. I feel like my maths degree does not help at all to approach truth. Maybe in these times, what we need in addition is just wisdom? But how can we find it (in a rational, not esoteric, sense)?

It often seems to me that these are really frustrating times for people honest enough to admit that they don't understand, while having been educated about, virtually anything in this world.

But given the success of Enlightenment, well-documented by Steven Pinker, I suspect that there have to be people with a clear-enough view of what's happening in the world to be able to improve standards of living for large parts of humanity, while managing nuclear deterrence and other dangerous constellations, it's probably just me who doesn't understand how all of this works.

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One doesn't need to assume an objective reality if one wants to be agentic. One can believe that 1) Stuff you do influences your prosperity 2) It is possible select for more prosperous influences.

The use of the concept of "effective" is a bit wonky there and the word seems to carry a lot of the meaning. What I know of in my memory "effective method" is a measure of what a computer or mathematician is able to unambigiously specify. I have it hard to imagine to fairly judge a method to be ineffective.

Just because you need to have a starting point doesn't mean that your approach needs to be axiomatic.

It is unclear why planetary consiouness would be desirable. If you admit that you can't know what happen on the other side of the planet to a great degree you don't have to rely on unreliable data mediums. Typically your life happens here and not there. And even if "there" is relevant to your life it usually has an intermediary through which it affects stuff "here".

I'm very grateful that you bring up these points. Sorry for the long response, but I like your comment and would like to write down some thoughts on each part of it.

One doesn't need to assume an objective reality if one wants to be agentic. One can believe that 1) Stuff you do influences your prosperity 2) It is possible select for more prosperous influences.

First of all, I think choosing the term "objective" in my post was a too strong, and not quite well-defined. (My post also seems at risk of circular reasoning because it somehow tries to argue for rationality using rationality.)
I really should have thought more about this paragraph. You proposed an alternative to the assumption of an objective reality. While this also requires the assumption that there are some "real" rules that tell which of one's actions cause which effects to my sensations today or in the future, and thus some form of reality, this reality indeed could be purely subjective in the sense that other "sentient beings" (if there are any) might not experience the same "reality", the same rules.

The use of the concept of "effective" is a bit wonky there and the word seems to carry a lot of the meaning. What I know of in my memory "effective method" is a measure of what a computer or mathematician is able to unambigiously specify. I have it hard to imagine to fairly judge a method to be ineffective.

What I mean by effectiveness is a "measure of completeness": If some method for obtaining knowledge does not obtain any knowledge at all, it is not effective at all; if it would be able to derive all true statements about the world, it would be very effective. Logic is a tool which just consists of putting existing statements together and yields new statements that are guaranteed to be true, given that the hypotheses were correct. So I'd argue that not having logic in one's toolbox is never an advantage with respect to effectiveness.

Just because you need to have a starting point doesn't mean that your approach needs to be axiomatic.

This is not clear to me. What do you think is the difference between an axiom and a starting point in epistemology?

It is unclear why planetary consiouness would be desirable. If you admit that you can't know what happen on the other side of the planet to a great degree you don't have to rely on unreliable data mediums. Typically your life happens here and not there. And even if "there" is relevant to your life it usually has an intermediary through which it affects stuff "here".

This is also a very good point, and I'll try to clarify. Consider an asteroid that is going to collide with Earth. There will be some point in the future where we know about the existence of the asteroid, even if only for a short time frame, depending on how deadly it is. But it can be hard to know the asteroid's position (or even existence) in advance, although this would be much more useful.

So, in a nutshell, I'm also interested in parts of reality that do not yet strongly interact with my environment but that might interact with my environment in the future. (Another reason might be ethical: We should know when somewhere on the world someone commits genocide so that we can use our impact to do something about it.)

So maybe the problem is a lag of feedback, or hidden complexity in the causal chain between one's own actions and the feedback, and this complexity requires one to have a deep understanding of something one cannot observe in a direct way.

With effectiveness my doubt is that you iss kinds of knowledge in your definition and that logic might be less than effective in the grander scheme of things. For example the knowledge of how to ride a bike is hard to get into the scope of logic, in that respect logic is incomplete ie it leaves a bit of knowledge out. There is the issue with Mary's room and whether color experience counts as knowledge, we can grant her allthe math test books and science books but we can still doubt whether we have caught all knowledge. Even the context of "effective method" Turing suspected that mathematicians use a kind of "insight" that coming up with a proof is a different kind of process than following a proof. Universal turing machine captures "effective method" which encompasses all of formal mathematics that person could write down. But still doubt lingers whether that is all the intersting kind of processes.

One could also be worried about a method of knowing that encapsulates logic. Divine relevation could be posited to give vast amounts of knowledge maybe enough so that further knowledge production work ceases to be viable. There is also the "trivial theory of arithmetic" where we just assume all arithmetic truths as axioms. In such a system there are no theorems, there is only a check whether or not a thing is a axiom or not. Such a system could be all encompassing and avoid the use of logical inference.

Starting point is a bit undefined, axiomaatic approach is way more defined. Sure we don'thave super cdertani "boot-system" on how we get going. But it doesn't feature the characteristics of a axiomatic system. In the axiomatic style you can go "Assume X. Is it the case that X?" and you can definetely that "yes, X is the case". If you tried to shoehorn the sensory reliance in axiomatic terms it would go something like "Assume X. Now turns out that X isn't the case" which is non-sense in proof terms. Sure there is appeal to absurdity "Entertain subthought:[Assume X. X leads to contradiction]. Because subthought is contradictory the axiom set can't all be true at the same time. Therefore not-X.". But when our sensory expectations are violated they are not appeals to absurdity, it is more of a trial and error of "Guess X. If X then Y is a prosperous choice. Experience of Y is very unprosperous. Regard X as bad guess.". A purely axiomatic approach will always refes to the starting definitions to resolve issues of truth. We don't need to guess our axioms because we assume them true, which in effect we define to be true. "Assume all Xs are Y", "well what if I find an X that isn't Y?", "then it is not an X, thefore you can't find an X that isn't Y"

I get that getting asteroided would be my business. But knowing what half of china is going to have for lunch tomorrow really isn't, I am fine not knowing that I am fine that I don't have control over that they can have their culinary autonomy. When you would scan for impact asteroids you would not generally scan all things in the same way, but focus on paths and locations that could contain dangerous elements which means giving more scrutinity to some and less to others. There is also the issue of balancing the prediction horizon over several threats. Do you want to spend time getting an addtiional decade advance warning on a collider asteroid or do you want to get another decade advance warning on climate disaster? Just because you can fret about or control something doesn't mean you should. And integrating garbage can be more dangerous than acknowledging that you don't know.

Regarding logic and methods of knowing, I agree that logic might not be the only useful way of knowledge production, but why shouldn't you have it in your toolbox? I'm just trying to argue that there's no reason for anyone to neglect logical arguments if they yield new knowledge.

I agree that "prior" is a vastly better word choice than "axiom" because it allows us to refine the prior later.

The "planetary consciousness" thing also appears to me as a misunderstanding: I don't want to propose that every information about the world should be retrieved and processed, in the same way that even in my direct environment, what my neighbour does in his house is none of my business.

How do you differentiate between "Truth" and "truth"? I would really appreciate some clarification regarding these two words because it would help me to understand your comment better. Thanks :)