## Is it possible to know anything with certainty?

7

1

I have been thinking about objectivism vs relativism recently.

It is easy to prove by contradiction that there exist objective truths. However, is it possible to know anything?

If you assume a human makes errors in logical deduction 5% of the time, then it seems to follow that it is impossible for a human to know anything (eg how to know 1+1=2). In which case it seems that the answer to my question is unknown. This is confusing to me, any thoughts?

Know this. In the relative blink of an eye, you'll be dead. We all will. Life is incredibly short, and utterly without any profound meaning. Love and honour your family, try to live with honour, and get drunk with friends when you can. – Richard – 2017-05-13T00:01:35.247

"It is easy to prove by contradiction that there exist objective truths." Can you give an example? Even "I think therefore I am" is a subjective truth. As for anything else, we could be a brain in a vat or the plaything of a Cartesian demon. – user4894 – 2017-05-13T03:45:21.670

A subtle distinction: are you seeking to know a truth, or a truth about reality. The former is easier as you can define the context. The latter is the subject of a great deal of philosophy. – Cort Ammon – 2017-05-16T16:28:34.130

7If the answer is no, then you know that with absolute certainty. Therefore, the answer is yes. – yters – 2014-08-14T18:55:59.330

@yters: You said "if" though. – TheDoctor – 2014-08-14T22:13:43.193

Please see the following for the rule of disjunction elimination: https://en.wikipedia.org/wiki/Disjunction_elimination

– yters – 2014-08-16T04:05:41.003

@yters: cute. dunno if it completely works. – robert bristow-johnson – 2015-06-18T22:09:49.927

It is certainly possible to be absolutely certain ... and absolutely wrong. GIGO. Correct operations upon mistaken premises do not generate useful results. 1+1=2 only if we agree on definitions of those symbols and their interpretation which make that a true statement; if we do, then it is true by definition within that system. We know that math is self-consistant, and we know it produces results which make useful predictions about the real world, and we find ways to refine it when if falls short (such as complex numbers)... that's as much truth as science ever offers. – keshlam – 2015-09-25T10:01:25.963

"No one can know anything", "One can know something", "I don't know anything" - all of those prove you know something. – None – 2016-01-04T20:21:27.673

I know people who know everything with certainty. Even if they are wrong. Which these people usually are. – gnasher729 – 2016-03-07T22:10:57.477

My father used to say to be definite no matter what. No one likes a person with no preference or who cannot take a position. Well, that is what he said, anyway. I don't have that position. – None – 2016-03-08T00:01:31.890

1

Yes it is through a process called "Deductive reasoning". Deductive reasoning implies that if all of the premises are true and if the inferences are valid, it follows that the conclusion must be true. Here's an example.

I have a bag full of black marbles. I will pull out a marble and record what colour it is until the bag is empty. It follows that I will only have recorded that there are black balls in the bag.

Let's break this argument down:

1. I have a bag full of black marbles.
2. I will pull out a marble and record what colour it is until the bag is empty.
3. I will only have recorded that there are black balls in the bag.

Given that premise (1) and (2) are true, it follows that the conclusion (3) must be true.

To answer your question, in the event of an argument that uses deductive reasoning, it is possible for something to be known in absolute certainty. This works in theory, however in practise it is harder to say that all given premises are true.

−1 You forget (!) that memory is fallible. You can't be sure that your memory of the colors of the marbles you have left, is true. In short, this answer is bollocks. – Cheers and hth. - Alf – 2017-05-23T05:39:36.383

4Yes, but suppose you made a mistake and incorrectly deduce something which is not true. I mean, if I knew everything which I deduced using deductive reasoning, then wouldn't I get 100% on all of my maths tests? – grasevski – 2013-09-23T04:10:02.123

As I said, in practise the lines of certainty become more blurred, deductive arguments are replaced with inductive arguments. – None – 2013-09-23T04:21:54.373

1Thank you for your help. What do you mean? If I claim 1+1=2, that would be proven using deductive reasoning, not inductive reasoning. I don't keep adding up 2 and average the results, I would start with some basic axioms and inference rules and then provide a proof using these rules that 1+1=2. This would be a deductive proof. My question is, how can I know that 1+1=2, given that I could make a mistake in my proof? And how then can I really know anything for that matter? – grasevski – 2013-09-23T04:44:41.943

In short, you can't. There isn't really a way that you can be completely, entirely, 100% sure without a shadow of doubt. In theory it's possible but in practise it just isn't possible. With my example I gave above, in theory it works but in practise someone could have slipped a red marble in the bag for example. – None – 2013-09-23T05:50:02.770

3This is only true if you can be 100% sure that deductive reasoning is capable of providing certainty about things, which we (quite clearly) cannot say. – stoicfury – 2013-09-23T08:31:35.977

It takes more than a hundred pages to prove 1+1=2 (The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913.) . see also http://math.stackexchange.com/questions/243049/how-do-i-convince-someone-that-11-2-may-not-necessarily-be-true

– MKR – 2015-06-17T07:28:30.847

4This answer is confusing the certainty of the deductive process with the certainty of the conclusion. A deductive argument may very well proceed from uncertain premises to an uncertain conclusion. It does not follow that a conclusion arrived at deductively is certainly true. – Bumble – 2016-01-04T20:27:24.400

@MKR It took more than a hundred pages to prove 1+1=2 in PM. It doesn't take nearly that long to prove it in, say, ZF. – Era – 2016-03-07T15:06:23.363

7

It is possible to know a very limited set of things with absolute certainty, i.e. not dependent on basic fundamental assumptions that "could" end up being wrong.

For instance, things I am absolutely certain about:

• I know with absolute certainty that I am not omniscient at this moment in time.
• I know with absolute certainty that my visual and tactile senses are telling me I am typing up a post on philosophy.stackexchange.com.

However, I do not have absolute certainty that I am actually typing this post in reality (this could be an elaborate simulation, I could be hallucinating while in a coma, etc). I also do not have absolute certainty that I was not omniscient 5 minutes ago, despite my memories telling me otherwise.

But if you are willing to move the line of certainty back from "absolute," the world becomes a lot more reasonable. Reasonable certainty is a lot easier to attain ;)

Interesting. I have few questions which, I'm afraid, might make little sense:

1. Are you absolutely certain that "you know with absolute certainty that you know with absolute certainty that you're not omniscient at this moment in time"?
2. Are you absolutely certain that "at this moment in time" is well defined?
3. Are you absolutely certain that your visual and tactile senses are as physically real (whatever that means) as you're assuming they are and that their physical interactions are the cause of your feeling (you might be a computer simulation)?
4. < – MMa – 2017-06-16T05:03:10.317

@Mohammed 1.) Yes; I, as a thinking agent, am without full and complete knowledge of everything. I don't even know your real name (assuming you exist at all). 2.) No. 3.) Nope, physical reality (and our senses' observations of that physical reality) is a basic assumption upon which I rely, but it is an assumption I will likely never be able to prove with absolutely certainty. – immortal squish – 2017-06-16T15:10:26.930

In my question, I was trying to see if you consider basic mathematical logic itself as absolutely certainly correct, rather than reasonably certainly correct. What is your stance on this? – MMa – 2017-06-16T17:46:04.947

That depends what you mean by basic mathematical logic. I am absolutely certain that 2 + 2 = 4, but that is because the logical construct of '2' is defined in such a way that it must result in '4' when added to itself. Beyond that, I am limited by my mathematical ability to understand and confirm what properties are necessarily true. – immortal squish – 2017-06-16T20:51:05.190

I'm trying to push you to handle "certainty" at a more fundamental level. It's so fundamental that it might be boring to you or downright pointless. But it really bothers me. So let me try with the following. The reason you can feel and express your absolute certainty about basic mathematical statements like 2+2=4 (but, say, a computer can't) is because you have consciousness (I want to guess, do you agree?). But the basic fact that humanity isn't even close to understanding consciousness at a fundamental level leaves no room for absolute certainty about literally anything. – MMa – 2017-06-17T06:26:00.027

Certainty is a subjective measure of confidence, so I agree consciousness appears to be necessary. I disagree that one needs complete understanding of a thing to have absolutely confidence in any aspect of that thing, e.g. I can have absolute confidence in the logical absolutes, without understanding the fundamental nature of reality (and why those logical absolutes are consistently reliable). – immortal squish – 2017-06-19T16:08:57.180

If the nature of consciousness or how you acquired it isn't something you're absolutely certainly about then the (ridiculous) possibility that an agent designed it for you with the intention of making you feel confident about the validity of any statement of mathematics couldn't be eliminated with absolute certainty. – MMa – 2017-06-19T19:35:29.130

Sure, but regardless of how my consciousness came about, it is currently a thinking agent with sensory & knowledge limitations. Even if everything I am experiencing is a fabrication, and my consciousness is only temporarily segregated from some omniscient/omnipotent whole... my previous statements on absolute certainty still hold true for this current situation. – immortal squish – 2017-06-19T19:56:23.640

3

## TL;DR

If to know something with certainty means having undoubtable, true thoughts, the answer is: We cannot even determine for certain whether we know anything about the world [i.e. anything that we learn through our senses], but we can know the form of our thinking (and sensing, and other faculties) for certain.

## His argument

### What is truth?

The nominal definition of truth [...] is the agreement of cognition with its object [die Übereinstimmung der Vorstellung mit ihrem Gegenstand]. (CPR A58|B82)

### The problem

...but one demands to know what is the general and certain criterion of the truth of any cognition (ibid)

Without a criterion of truth that can be applied to all cognitions, it is hard to determine the truth value of a cognition, as we would need to do so through cognitions of the object...and these have to be true.

We have the problem of getting into an infinite regress here, because all we can know we know through cognition.

### First step: We need a determinate object

If truth consists in the agreement of a cognition with its object, then this object must thereby be distinguished from others; for a cognition is false if it does not agree with the object to which it is related even if it contains something that could well be valid of other objects. (A58|B83)

The cognition must contain something that is valid only for the object in question and not for other objects, i.e. it must be determined and distinguishable.

### Step two: A criterion of truth must be valid for all objects of cognition

Now a general criterion of truth would be that which was valid of all cognitions without any distinction among their objects. (ibid)

The criterion/property of truth must be applicable to all objects.

### Step three: Incoherence between above definition of truth and necessary properties of a criterion of truth

But it is clear that since with such a criterion one abstracts from all content of cognition (relation to its object), yet truth concerns precisely this content, it would be completely impossible and absurd to ask for a mark of the truth of this content of cognition, and thus it is clear that a sufficient and yet at the same time general sign of truth cannot possibly be provided. (A59-60|B83)

This one needs some explanation. 1) We saw above that truth is defined in relation to the object of cognition (i.e. correspondence). 2) If a criterion for truth should be applicable to all cognitions, it has to abstract from any particular object. 3) Without referring to a particular object, there is no sufficient way to determine truth 4) A criterion of truth is - by definition - either not sufficient or not general.

### Conclusion: A general criterion of truth is self-contradictory (and all criteria of truth about empirical things are therefore arbitrary).

Since above we have called the content of a cognition its matter, one must therefore say that no general sign of the truth of the matter of cognition can be demanded, because it is self-contradictory. (A59|B83)

So if we speak of any criterion for the truth of the matter of cognition (i.e. anything that is given by sensibility in the form of (empirical) intuition, see A50-52|B74-76), the answer has to be that there cannot be certainty, as truth value will always be contingent, depending on empirical habits and findings.

### Last step: But there is hope! The form of thinking is always the same, independent from the actual matter of cognition!

But concerning the mere form of cognition (setting aside all content), it is equally clear that a logic, so far as it expounds the general and necessary rules of understanding, must present criteria of truth in these very rules. For that which contradicts these is false, since the understanding thereby contradicts its general rules of thinking and thus contradicts itself. But these criteria concern only the form of truth, i.e., of thinking in general, and are to that extent entirely correct but not sufficient. For although a cognition may bc in complete accord With logical form, i.e., not contradict itself, yet it can still always contradict the object. (A59|B84)

Well, to be fair, there is this big caveat: This means that "truth" is reduced to "it is a necessary truth that incoherent cognitions and invalid reasoning cannot be true", i.e. logics is a conditio sine qua non (ibid), but does not help us to positively determine the truth value of any cognition (A60|B84)

### Aside

Apart from the form of thinking (or, more clearly in that context: discursive judgeing), Kant found what he thought to be necessary truths in the form of sensibility (CPR, transcendental aesthetics), cognition (CPR, transcendental logics), moral judgements (CprR) and judgements in general (CPJ). The finding and deducting (i.e. justification of its validity) of these propositions or synthetic sentences a priori is the aim of his critical philosophy.

### Abbreviations

CPR - Critique of Pure Reason (A edition 1781, B edition 1787) CPrR - Critique of Practical Reason (1788) CPJ - Critique of the Power of Judgement (1790)

2

Okay, I am someone who does not come from a philosophical background, so I don't really have credentials and I can't really give you any nice sounding quotes from philosophers but I do think that I can help you answer that question for yourself. But I can help you out from the perspective of a computer scientist, and while that seems unrelated, I'll show you how it's actually very relevant.

You see, we computer scientists need to be able to reason about the outside world, and to do that, we come up with models. Models are nice because they are finite and you can not only reason, but you can also prove hypotheses. For example in a model of the traffic lights of an intersection, the traffic light can only be green or yellow or red, and not all at the same time, or dim red, or blue, because you define the model that way.

You can make models in mathematics, for example. If you take all natural numbers, define the + operator to mean your usual addition, then you can prove that 1 + 1 = 2 definitively. However, there is nothing preventing you from defining the model so that 2 + 2 = 5, it would just be a rather useless model.

Now, the reason we need models is because it's impossible to reason about the real world. And there are two ways I can illustrate this.

Firstly, there is language. If you dig a little bit into linguistics, which I as a computer scientist am familiar with because trying to understand and generate natural language is a problem we've been working on for a while, you'll see that the biggest problem with human language is that it's ambiguous. And since our natural language is the only tool we have for reasoning about the real world, we can't really have absolute truths. And for that same reason, it's so hard for a computer to understand natural language. Since I'm not a linguist, I unfortunately can't really go much into detail explaining why language is ambiguous, but think about this: if I say, "There is a cup on the table," then that is a very ambiguous statement. What is a cup? Is it a cup because it's atoms are arranged in a special way? Is it a cup because you use it like a cup? Is it a cup because it has a handle that looks a certain way? Would you still call it a cup if it were made out of a radioactive material? Would you still call it a cup if it had a hole in the bottom? Is it still a cup if it didn't have a handle? You see, the word "cup" is not well-defined, it's another model.

To be able to have absolute truths, we'd need to have absolute knowledge, and we are unable to get that because we simply don't have the mental capacity. I dare say, to have absolute truths about the universe, we'd need to know the position and velocity of every single atom, electron, neutron, positron, neutrino, quark, photon and whatever news things the physicists discover, and to hold that information we'd need a brain that is large enough, and to have a brain that is large enough, it'd need to be larger than the universe because we need more than one atom to store a bit of information, and with a brain that size, the three-dimensional interconnections would be too slow, so we'd ideally need to live in a higher dimension as well.

My point is, our perception is limited by what we can see (two-dimensional image of a narrow range of photons), hear (short range acoustical signals), feel, smell, and process (with our inherently ambiguous and limited language). So, in effect we are limited to reasoning about models that we make of the world. That is how most of science works anyways, and the goal is to expand the model to make it as close to the real world as possible, while still allowing us to reason properly.

Welcome to Philosophy.SE. If you would like, here is a link to take the tour. Thanks for your participation.

– PV22 – 2017-05-27T19:00:51.140

2

Yes it is possible to know things with absolute certainty, however, it is also possible for someone else to claim to know the negation of it with absolute certainty.

Some examples, commonly referred to as necessary truths:

All things are selfidentical.

There is no thing such that it is a circle and it is squared.

There is no thing such that is is a horse and not a horse.

1+1=2

1

Yes it is possible, but then that part of your life becomes dependent on that assertion. This is why there are axioms and logic which survive over time: they simply become the status quo which is defended for various reasons: ultimately for either practicality or aesthetics.

Beyond that one can only know of one's existence. Because to know anything means to exist. That is the starting point, the basis for philosophy.

1

It is easy to prove by contradiction that there exist objective truths. However, is it possible to know anything?

I believe this is where the ugly but necessary bridge between Continental Philosophy and Analytic Philosophy is very necessary. Although, given a set of axioms, prove rigorously the existence of certain truths, we explicitly rely upon the ontologies presented within the axioms in order to make any claims to truthfulness. Take for example what @Eliran writes:

I have a bag full of black marbles.

How do we know that we have this bag? We rely explicitly upon a world of phenomena, limited to our own perception, and uncertain of the extension of that perception to any sense of universality. There is an implicit ontology of "having", related ultimately to the "being" of these black marbles. There is a Cartesian instability to that existence, but we cannot rely on God's willing hand to move things back into place like Descartes did. Rather, we must move to the assumption that the bag does exist, but keep in mind that this assumption works not in any universal sense but rather in our own "lifeworld" (this is from Husserl's Crisis of the European Sciences). We cannot know that these black marbles exist in any universal sense, but we can observe that in our world, that they do exist.

This is not relativism - this is bracketing universality not to the world, but to all experiences of the world. Thus, that we draw only black marbles does not serve as a universal truth (because can we know with certainty of the a priori world outside of our experience?), but rather, as a norm. This is kind of leading all up to Habermasian communicative rationality, but I'll leave you to research the topic (though I don't recommend it particularly as a model for political-moral norms, it works perfectly well as a model for scientific rationality).

If you assume a human makes errors in logical deduction 5% of the time, then it seems to follow that it is impossible for a human to know anything (eg how to know 1+1=2). In which case it seems that the answer to my question is unknown. This is confusing to me, any thoughts?

Why does the human make these errors in logical deduction? Is it not simple, if given a set of axioms, that the conclusions may follow beautifully into place? The assumption of the "imperfect human" (imperfect in that they cannot calculate 1+1=2) is somewhat facetious - it's really that the human's (lack of) experience of the truthfulness of a validity claim (1+1=2) that can bring them to an incorrect conclusion.

0

You write:

If you assume a human makes errors in logical deduction 5% of the time, then it seems to follow that it is impossible for a human to know anything (eg how to know 1+1=2). In which case it seems that the answer to my question is unknown. This is confusing to me, any thoughts?

People don't make errors 5% of the time. On any particular occasion a person either makes an error or doesn't make an error. And errors arise from false ideas about what's going on or from mistakes about some particular fact. They don't arise randomly and don't arise just from deduction.

A better question is "How is it possible to create knowledge given that we might be mistaken about anything we think is true?" Any idea could be wrong and so we should be open to reconsidering any idea and should actively look for errors. Knowledge is information that solves problems. Knowledge is not guaranteed to be true or probably true or anything like that. We create knowledge by conjecture and criticism. We notice a problem, something about our current ideas that seems worth fixing. We then guess about how to fix the problem. We subject the guesses to criticism. Does the guess solve the problem we intended to solve? Does it conflict with other ideas and if so should the new idea or the old one be discarded? Does it conflict with experimental results? You keep criticising the guesses until only is left and then look for problems with that guess.

See "Realism and the Aim of Science" Chapter I by Karl Popper and

www.fallibleideas.com.

0

You can never be 100% sure that your memory of your thoughts or other happenings 1 second ago are true or completely accurate. For that matter the local part of the Universe might be a statistical fluke that popped into existence 1 millisecond ago, so that your apparent memories etc. are just arbitrary, consistent only to the very shallow degree that you're able to inspect them over such short a time interval. The chance of something like this happening is fantastically low, but not zero, hence you can't ever be absolutely sure about anything.

My argument is as follows:

1. Pick any short (apparent) claim that you thought needs a reference, e.g. that you can't be sure of your memory.
2. Now try to prove the opposite, that you can be absolutely sure of your memory.
3. You will not be able to do so.
4. Hence you can't be absolutely sure of anything.

@Keelan, Subordinating argument to narratives, pedigrees, names, histories, and warring tree-graphs of timestamps, expresses more love of opinion than knowledge. Collectors love their price guides, so perhaps the PSE point system fulfills them. – agc – 2017-05-22T22:22:51.170

1Without references, this is just giving your opinion. Please provide some references. – None – 2015-06-20T07:22:12.930

1There are many websites where random people can give their opinions about whatever questions people may have. We're trying to do something different here, by not discussing random people's opinions here, but by discussing those ideas that were brought forward by the study of philosophy through the ages. In that light, an answer that doesn't provide any references isn't very useful, because it doesn't help someone in studying philosophy. If you'd provide references to philosophers who claim this, one could look it up, search for antitheses, etc., that would be much more helpful for his study. – None – 2015-06-20T12:22:08.690

1@Keelan: The question as stated is not about history of philosophy, sorry. As I understand it you don't like an answer, because you want a study of the history of philosophy – Cheers and hth. - Alf – 2015-06-20T12:29:40.520

2Presuming we are still on philosophy.SE, the question should be about philosophy in some respect where philosophy is understood as it is defined generously around the contours of the historical discipline. – virmaior – 2015-06-20T16:47:07.457

I've deleted sections of comments that were pointlessly condescending... For reference ,the discussion of scope should take place on meta not here, and the seemingly inevitable cry of what me? about rude comments also belongs on meta -- not here. – virmaior – 2015-06-20T16:53:29.390

0

No. In the Bayesian interpretation of probability, probability represents subjective belief. So there may be an ideal world with a perfect mathematician can make a deduction, and be 100% guaranteed to be correct. But a bayesian reasoner in the real world must consider the possibility of errors, deception, or even crazy hypotheses like false memories. It must attach probabilities to all beliefs, and consider all hypotheses.

Humans are certainly not perfect. We make mistakes all the time. If nothing else, the neurons that run your brain are somewhat random and probabilistic, and you can't ever be 100% sure that your thoughts are memories are correct.

Can you really be sure about things in mathematics? Well published mathematical research is shown to have errors all the time. Everything is probabilistic. Nothing, not even math, can claim to be infinite certainty. How certain are you that 51 is a prime number?

0

No it is not possible. My argument.

P1: If, we can claim that things in the universe are a certain way with absolute certainty, then we must have complete knowledge of the thing

P2: If we have complete knowledge of a thing, we have to have complete knowledge of the universe in order to understand the thing completely.

P3: We do not have complete knowledge of the universe.

P4: We can not claim things in the universe are a certain way.

Does you argument apply only to physical reality or even to mathematical logic? – MMa – 2017-06-16T04:48:44.903

0

All proofs or arguments (deductive or otherwise) are finite; if they weren't, the conclusion could never be reached. They rely on premises, and those premises, forming the basis for the argument, are unproven.

You could create a proof of those premises; but that new proof would in turn itself rely on unproven premises. So the problem is inescapable. Ultimately, all deductive reasoning depends on premises that aren't deductively proven.

Arguments often also rely on auxiliary premises that aren't explicitly present. They are background assumptions.

Premises fall into three categories: arbitrary premises, like those chosen for an abstract formal system; provisional or working premises; and self-evident truths.

Since arbitrary premises are stipulated, and provisional premises are uncertain, we can move on to the directly apprehended, or self-evident, truth. This could be something experienced, like consciousness, or it could be something relying on intuition or some other form of understanding.

Sometimes, however, things taken to be "obvious" or self-evident are dependent on auxiliary premises that needn't be true, or are otherwise only provisionally true. Provisional truths can be changed to provisional falsehoods (and vice-versa) by additional data, for example.

The problem is complicated by two additional, often overlooked factors: the circularity of definition; and the fact that statements only have truth values if they are sufficiently well-defined.

In order for a statement to be well-defined, each individual term must be well-defined. (Also, of course, the words must be put together in a meaningful way.)

Every dictionary, no matter how technical or comprehensive, contains a finite number of terms. So, every definition, using terms in the dictionary, must point to other terms in the dictionary, which in turn point to other terms in the dictionary. Eventually, some of the definitions must point back, circularly to terms (original or intermediate) that one is trying to define.

So, meaning can't come from from​ formal definition without circularity. The true meaning comes from personal associations or apprehensions attached to terms. For example, "red" means something to me not because it is defined as a mixture of other colors or as certain wavelengths of light, but because I have seen objects to which the term has been applied.

So, already, the concept of rigor is starting to get fuzzy, or at least subjective.

As for truth values, they can't be assigned to obvious nonsense. "Bling fort shnozzle" isn't true or false, it's simply gibberish.

But what about statements that seem to be talking about something, but involve fuzzy conceptualizations or poorly defined terms, or contradictory elements?

Geometric points are said to have no spatial extension yet somehow occupy and indicate position within space.

Real numbers contain a never-ending series of digits that are nevertheless said to form completed entities in their "entirety".

Deterministic processes are said to be ultimately the result of random quantum events.

"Two things can touch one another." What could be simpler, or clearer?

If there is space between them they aren't touching. And if there isn't space between them they must be spatially coincident, which is to say they must partially occupy the same space simultaneously. Except that because of quantum fuzziness, objects made of atoms with orbiting electrons don't have clear borders. And because of special relativity, simultaneity means something different to observers in different frames. And time is subject to quantum uncertainties. And mathematically precise position ultimately comes down to geometric points, which are nothings pretending to be somethings.

I think those concepts involving 'contradictory elements' is just caused by the vagueness and ambiguity of natural language. As Wittgenstein said, natural language has a lot of defects. Maths, philosophy and sciences are all just games of languages, they're using their own language systems instead of the unprecise natural language. Natural language, and even all languages, are never talking about what the true nature is. About the three concepts you mentioned (points, real numbers, random events) I also have some own understandings. If you're interested you can contact me. – Antimonius – 2021-01-21T03:08:38.050