There's a religious zeal associated with the mantra "Doubt Everything". Doubt is not a bad thing, provided it does not dissuade you from doing mathematics or playing chess or squeezing lemon onto your fish and chips.

Those who follow Wittgenstein hereabouts, myself included, have been a little bit disingenuous in their reporting of OC, as those who have actually read it will know. Time to come clean.

The indubitable can indeed be doubted, albeit only under specific circumstances.

The argument is, roughly, that in a given language game (and it is all language games), there are certain things that cannot sensibly be doubted. So in geometry the three angles of a triangle add to a straight angle and in Chess the bishop moves only diagonally.

However, language games themselves are subject to change. So in some geometries the angles of a triangle add to more than a straight angle, in others to less; once the pawn could only move one square, but to speed the game this was changed to two squares for its initial move.

In such cases it is very important to understand which game is being played.

There's a religious zeal associated with the mantra "Doubt Everything". Doubt is not a bad thing, provided it does not dissuade you from doing mathematics or playing chess or squeezing lemon onto your fish and chips.

Those who follow Wittgenstein hereabouts, myself included, have been a little bit disingenuous in their reporting of OC, as those who have actually read it will know. Time to come clean.

The indubitable can indeed be doubted, albeit only under specific circumstances.

The argument is, roughly, that in a given language game (and it is all language games), there are certain things that cannot sensibly be doubted. So in geometry the three angles of a triangle add to a straight angle and in Chess the bishop moves only diagonally.

However, language games themselves are subject to change. So in some geometries the angles of a triangle add to more than a straight angle, in others to less; once the pawn could only move one square, but to speed the game this was changed to two squares for its initial move.

In such cases it is very important to understand which game is being played. — Banno

MU, and I assume also Magnus, are using "doubt" to mean "lack of certainty", and it seems reasonable to lack certainty if something isn't certain.

If I know that something could be false then I know that it being true isn't certain, and if it being true isn't certain then it would be unreasonable to be certain that it is true.

It seems to me (although correct me if I'm wrong) that you're misinterpreting them as saying that they have a reason to believe that it is false (as opposed to just could be false), or at least that the evidence that it's true isn't compelling. But that doesn't seem to me to be what they're saying at all. They're just saying that because it isn't certain, they're not certain. What's wrong with that?

Certain means the same as necessary?

Certain means the same as necessary? — Banno

Or just 100% probability, if that's different.

Oh, yes.

Necessary implies it is true in every possible world.
100% probable implies it is true.
Being certain implies that it is indubitable.

Three distinct things, unless someone can show that they are the same.

I've always liked Adorno's quip that those who fetishize doubt always ultimately end up doing so in order to better secure certainty, rather than for any intrinsic affinity for doubt as such: that absolute doubt and absolute certainty, far from being opposed, always end up walking hand in hand. I think he's mostly right about this, and speaks to one of the reasons Cartesian doubt has always struck me as a kind of hilarious false drama, as if a drama queen were to write philosophy.

Cartesian doubt has always struck me as a kind of hilarious false drama, as if a drama queen were to write philosophy. — StreetlightX

Indeed; even Descartes thought so.

But even he ended up taking himself seriously in the end - and inspiring legions of lost souls in his wake.

What do you mean by "actually given in perception"? How could this be different than what people in general consider to be given in perception? — Janus

There must be a distinction between what I think is given in perception and what is actually given in perception, just as there is a distinction between what I think is on the menu at Sizzler and what is on the menu at Sizzler. What is actually given in perception is what you actually see. What is 'there' before your conscious awareness. What I think is given may be utterly different.

Still, I think we might agree. I say that perceiving a hand is sufficient to end a regress of sceptical questions. You say that the ostensive definition 'this is a hand' is sufficient. I am not sure the two views are really different.

Cartesian doubt has always struck me as a kind of hilarious false drama, as if a drama queen were to write philosophy. — StreetlightX

Descartes' announces the aim of his doubts in the opening lines of the Meditations, and so I find it incredible that people have so often failed to understand his purposes. To me, he is perfectly clear. He was trying to find something 'stable and lasting'. He wanted to give an argument for some of his opinions - an argument so strong that it couldn't later be over turned. It isn't clear to me that this necessarily meant solving the regress problem or engaging in some sort of false drama. If you want to give a great advantage to your own philosophical theories over others, and you have just seen the fall of the most respected system so far (from Descartes' perspective) - the Aristotelean one - then what he tries to do makes perfect sense.

Best,
PA

Right, so, as I said - absolute doubt goes hand in hand with absolute certainty (that which is 'stable and lasting'). It's as if trying to found philosophy on pathology (I don't even mean this polemically - there have been multiple readings of Descartes as the first to introduce madness proper into our conception of thought, which is in a way a kind of advance in the history of philosophy, if only Descartes didn't paper over it almost immediately). It absolutely makes sense. It's also awful.

What is pathological about desiring one's theories to be stable and lasting - immune from future upheaval? What is awful about what Descartes does?

No as in, Descartes quite literally introduces madness into thought in the form of the malin génie, the 'evil demon' which signifies the utmost derangement that Descartes can imagine. Pathology is at the heart of Descartes 'rationalist' operation, it founds his entire method of approach, not even implicitly but explicitly - he is all too happy to actually do this on his own terms.

2+2=4 is not immune to doubt? But doubt here could only mean that the doubter did not know what "2", "+", "=" or "4" meant... — Banno

Why would you say that doubting the meaning of something is not a sensible form of doubt?. I think not understanding the meaning is the basis of all doubt. As I described earlier in the thread, those who doubt the existence of the external world do so because they doubt the meaning of "existence". Since it has not been demonstrated to them what it means to exist, such that "existence" could be applied to external things, they doubt whether "existence" can properly refer to external things. So they are completely unsure (doubtful) as to whether the external world has any existence, because "existence" has been assigned to it, but what it means to exist has not been explained, demonstrated, or justified.

So what is it they are doubting? Not that 2+2=4, because they do not understand what that means, and so could not doubt it. — Banno

Why do you think that one could not doubt what they do not understand? Isn't "not understanding" the very cause of doubt, just like understanding is the cause of certitude? Someone sees these symbols "2+2=4'", and recognizes that they are symbols, but has doubt, because the meaning is not known

The argument is, roughly, that in a given language game (and it is all language games), there are certain things that cannot sensibly be doubted. So in geometry the three angles of a triangle add to a straight angle and in Chess the bishop moves only diagonally. — Banno

If a person does not understand the rules of a particular game, then the person looks at those rules with doubt. This is completely sensible. What doesn't make sense is to assert that those rules cannot sensibly be doubted.

So the tyrant dictates: "these are the rules and you cannot sensibly doubt them, because you are my subjects, and you have no choice but to play my game". Nobody expects the Spanish Inquisition.

2+2=4 is not immune to doubt? But doubt here could only mean that the doubter did not know what "2", "+", "=" or "4" meant...

So what is it they are doubting? Not that 2+2=4, because they do not understand what that means, and so could not doubt it. — Banno

Mathematical equations such as 2+2=4 are not immune to doubt. They can turn out to be wrong. One only has to understand how.

How do we determine whether any given mathematical equation is true or false? This is the question we must answer.

Any given mathematical equation is true if it belongs to the set of all valid mathematical equations. We can narrow this down by saying: any given mathematical equation of the form a + b = c is true if it belongs to the set of all valid mathematical equations of the form a + b = c. This is still complex. We need something simpler. Let's focus on the logical operation of negation. Any logical expression of the form not p = q is true if it belongs to the set of all valid logical expression of the form not p = q. Still, this is somewhat complex. To make it simpler, I'll generalize it. Instead of speaking of a specific logical expression that is negation I'll speak of a unary operation on a set of bits. Thus, any given unary operation on a set of bits, op x = y, is true if it belongs to the set of all valid unary operations a on set of bits. This will allow me to escape social conventions and use language any way I want. It will allow me to demonstrate that 2 + 2 = 4 is true not because of social conventions but because of what the individual decides to be the set of all valid mathematical equations.

So what is the set of all valid unary operations on a set of two bits? You choose. It's a personal choice. It can be anything you want. For example, it can be {(1,0), (0,1)}. This would be what most call negation. But you don't have to call it that. You can call it anything you want. You can call it "fuck your mother bastard" if you are badass enough. What is important is that the individual himself chooses the set against which he's going to be comparing mathematical expressions for their validity.

My orientation is extensional rather than intensional. I focus on actions first and words second. I have a negative opinion of philosophies that put way too much emphasis on language and other social conventions.

The set of all unary operations on the set of bits {0, 1} is {(0,0), (0,1), (1,0), (1,1)}. Now we have to choose the set of all valid unary operations on a set of bits. Let this be {(0,1), (1,0)}. This means that unary operations such as (0,0) and (1,1) are invalid (a.k.a. false) and unary operations such as (0,1) and (1,0) are valid (a.k.a. true.)

The only thing left to do right now is to explain how it is possible to be wrong about these sorts of statements. How is it possible to be wrong that 2 + 2 = 4?

There are three things we must focus on:

1. the set of all valid mathematical equations
2. the mathematical equation under consideration
3. our judgment as to whether the mathematical equation under our consideration (2) belongs to the set of all valid mathematical equations (1) expressed as either true or false

It looks sort of like a deductive argument, doesn't it? Let's give an example.

1. the set of all valid ordered pairs of bits is {(1,0), (0,1)}
2. the order pair of bits under our consideration is (0,0)
3. the statement that the order pair of bits under our consideration (2) belongs to the set of all valid ordered pairs of bits (1) is true

But this is wrong, isn't it? It is not true that (0,0) belongs to the set {(1,0), (0,1)}. We made a mistake and this mistake has nothing to with language i.e. we did not make it because we failed to understand the concepts.

That was a simple example. In reality, we rarely make mistakes with simple calculations such as "not F = q". But when a calculation is sufficiently complex, mistakes of this kind are very common.

For example, a mathematical equation such as (235110 * 2 - 65261 + 81) * 163 - 1684 = 66019836 is more difficult to verify. If an average person was asked to calculate the result of (235110 * 2 - 65261 + 81) * 163 - 1684 it wouldn't be surprising if they made a mistake. And such a mistake, you will agree, has nothing to do with an inability to understand concepts. Most people who make such mistakes understand the concepts very well. More often than not, the cause of such mistakes is a weak concentration.

There's a religious zeal associated with the mantra "Doubt Everything". Doubt is not a bad thing, provided it does not dissuade you from doing mathematics or playing chess or squeezing lemon onto your fish and chips. — Banno

I am not saying "doubt everything". What I am saying is that "everything can turn out to be wrong". There is no statement that is indubitable. Rather, there are simply statements that we choose not to doubt for one reason or another. A man who doubts everything cannot act for action requires that he settles on what's going to happen. If I can't decide whether it is raining outside or not I can neither take an umbrella and go outside nor go outside without taking an umbrella with me. Most of us don't want to be stuck in a limbo, so we are more than happy to put an end to doubt. But that does not mean we stop doubting when we reach absolute truth. If I decide that it is raining and that I should take an umbrella with me that does not mean it is raining outside. It simply means I got tired of doubting which forced me to go with my best guess. I am not a fan of extreme skepticism but at the same time I am not a fan of dogmatism according to which there are indubitable statements. There is no such a thing as indubitable statements. Everything can turn out to be wrong including 2 + 2 = 4. 2 + 2 = 4 will turn out to be wrong when we realize, if we ever realize, that the set of valid mathematical equations that is of interest to us does not contain this expression.

There is no statement that is indubitable. — Magnus Anderson

Including this one?

My orientation is extensional rather than intensional. I focus on actions first and words second. I have a negative opinion of philosophies that put way too much emphasis on language and other social conventions. — Magnus Anderson

Wittgenstein might have said the very same thing. He was a senior engineer in the Austrian army during the war, and had a famous "negative opinion" of philosophy. The premise of the Investigations is that philosophers pay way too much emphasis on language, detaching it from the actions in which they gain their meaning; hence "the meaning of a word is it's use in a language game".

I doubt it. O:)

So have I understood you correctly? You are saying that because one might doubt more complex equations, one might also doubt that twice two is four?

@Michael -Did we reach any agreement on certainty, truth and necessity being distinct things?

What do others think of this?

Well, we can certainly agree that truth is distinct from certainty, given that the main disagreement is over whether or not we can be certain of anything, despite the fact that we likely agree that lots of things are true.

As for certainty and necessity, I suppose that depends on how liberal we're being with the term "necessity". If I'm a man and all men are mortal, is it necessary that I'm mortal? It necessarily follows from the premises, but there could a possible world where I'm not mortal (either because I'm not a man or because not all men are mortal).

Although I'm not really sure how this addresses the key claims that 1) if something is possibly false then it isn't certain, that 2) if something isn't certain then it is unreasonable to be certain of it, and that 3) to doubt is to lack certainty.

So have I understood you correctly? You are saying that because one might doubt more complex equations, one might also doubt that twice two is four? — Banno

You verify that a mathematical equation is true by checking whether it belongs to some set of mathematical equations. You verify that "not T = F" is true by checking whether it belongs to some set of mathematical equations. If this set is {"not T = F", "not F = T"} then it is true; otherwise, it might be not. This set has a location. This means you have to properly locate it. If you don't then your conclusions might not be true. If instead of locating {"not T = F", "not F = T"} you locate a set such as {"not T = T", "not F = F"} then you will conclude, erroneously, that "not T = F" is not true. And it is possible that it is aliens who are responsible for your failure to properly locate the set you want to locate.

You can think of this set in more concrete terms. Think of a drawer that contains balls that are equal in all respects except in color. They can be equal in color but not necessarily. Say you're holding a red ball in your hand and you want to know whether there is such a ball in the drawer. What do you have to do in order to find your answer? You have to open the drawer and look for a ball that is of the same color. But you might end up opening the wrong drawer . . . and this could be because of the aliens. Through some action at a distance, each time you try to open the drawer, these evil aliens replace the original drawer with a different one.

Do you get my point?

You verify that a mathematical equation is true by checking whether it belongs to some set of mathematical equations. — Magnus Anderson

I don't. I verify that twice two is four... well, I don't understand how it needs verification. In understanding what is being said, I am certain of it's truth.

It could be dubitable. But first, you have to give me an example of what it means to doubt such a statement. Apparently, it should mean that there are indubitable statements. But what would be the definition of an indubitable statement?

It's like when someone says "truth is there is no truth" and then some moron comes along and objects with a statement such as "but that would be a truth, no? you said it yourself . . . it's a truth which states there are no truths . . . . so your statement is self-defeating" which indicates nothing but autism on their part.

1) if something is possibly false then it isn't certain — Michael

So in your view only necessary statements can be certain?

Well, I don't think necessary and certain mean the very same thing. Because while I am certain that I am writing this sentence in English, in some possible world I am writing it in French.

No, certainty is a propositional attitude; necessity isn't.

In order for a statement to be testable (verifiable/falsifiable) there must be something against which it can be compared. Mathematical equations must be compared against something; otherwise, 2 + 2 = 5 is just as good as 2 + 2 = 4. There must be some sort of standard. If I want to speak in English I can't use any kind of words I want. I must respect the rules of English language.

No, certainty is a propositional attitude — Banno

It's not just a propositional attitude. If I remove 13 hearts from an ordinary deck of cards then it is certain that I won't find another, even if I believe otherwise. Or if I believe that there are only 12 hearts and so after removing 12 of them am certain that I won't find another than my certainty is misplaced, given that there certainly will be another.

Popper never intended falsification to be used on mathematics.

How is that an example of certainty not being a propositional attitude?

Michael is certain that there are no hearts in the pack.

Looks like it sets out your attitude to a proposition...?

Noone cares about Popper. Every test either verifies or falsifies that which is being tested. Popper had this weird obsession with falsification. Anyways, every analytical statement can be tested. One only has to understand that analytical statements are tested against some set of rules that determine what is allowed and what is forbidden. 2 + 2 = 4 can be tested against any set of definitions of concepts "2", "+", "=" and "4".