[ Epistemic status: way more tentative than it sounds. Talking out my hat. ]
As I made a point of in my other post, it is a simple bivalent truth that definability is beyond computability and as humans can define things that cannot be computed, they do not work solely computationally.
Wait. Definability is beyond computability. Gotcha. But I think there’s a reference problem when you say humans can do “it”. Humans can’t actually solve any of these problems. They can define them. I don’t know how much exploration have been done about whether definitions are computable, and whether a theoretical Turing machine can define problems that it can’t solve.
I suspect the answer is “yes”, definitions can be computable even if the thing defined is not. Which then squares my intuition that humans are actually computing devices, just very complicated ones.
I also object to the sloppy title—“choose one” has a pretty open meaning, and “one” is an absolutely valid choice. It was performed by my brain (or by the section of the universe that computes my thinking, if you prefer), so it was done computationally. Further, obXKCD: https://xkcd.com/221/
>But I think there’s a reference problem when you say humans can do “it”. Humans can’t actually solve any of these problems
You got me here. Problem can be a “trick” term in this regard, as this is a very simple definition. I am just calling it problem to lure people into a computational logic to see for themselves that it makes no sense and is not how they address the question or instruction.🙂
In fact definition is also not necessarily the right term.
Internally I did uncomputationally deduce that X=1, because it is the first natural number and X can be 1, and hence X is 1 in this particular context of me actually writing this text. Of course one could deduce something else, but it is what I deduced.
Only afterwards did I define X=1, as I recognized in principle I could have deduced something else if my natural numbers were sorted differently, but I deduced based on logical order not geometric order.
>I don’t know how much exploration have been done about whether definitions are computable, and whether a theoretical Turing machine can define problems that it can’t solve.
They are not always (in bivalent logic), as that is contradictory (for example I can self-referentially state: “This definition is not computable and X=1”, which is a valid definition, and true by definition, as you can define something to not be computable, even though it is already not computable as a definition is not a computation (for example I could define that the busy beaver function is “really really really not computable”, even though it is already proven it is not computable).
You can define the busy beaver function, but you cannot compute it. The abstract set of Turing machines and their computations define the busy beaver function, but it cannot be computed by a Turing machine. That was proven by contradiction, as you can verify on the internet.
I guess in polyvalent logic you can through all terms together as you have infinite options what the terms can mean. Also a bit of a pitfall, as I think in general polyvalent logic might be be more valid, as it applies in more situations, but in mathematical reasoning bivalent logic is used for good reasons (like if 0s might or might not be 1 your numbers system becomes pretty complex pretty quickly).
>I also object to the sloppy title—“choose one” has a pretty open meaning, and “one” is an absolutely valid choice. It was performed by my brain (or by the section of the universe that computes my thinking, if you prefer), so it was done computationally
Choose one has a well defined meaning. You take one number of the set of all natural numbers, and put it on the right side of the equation. It is open, as the set of natural numbers is infinite.
That is inherent to the process, not sloppy in any way. I would say it is a bit sloppy to suggest otherwise but whatever, I think my term “problem” was also sloppy, so maybe let’s say instruction😉.
You said “one” is an absolutely valid choice, I unfortunately have to say it is literally not a valid choice as it a string not a natural number. It seems you just converted what I said into a string and outputted it back to me. That is a valid computation, it is however not a valid definition for X.
It seems just internally restated your assumption that your brain performs definitions computationally. The question is if that is true?
I would say per definition it is not true as definition and computation have a different semantic meaning, and are also different terms syntactically as definition!=computation. It might be point out the obvious, but although it is obvious we are not always conscious of it. We are not always conscious of any definitions which is maybe because definitions are not that important. Even just in math there is undefinability like the undefinable real numbers which make up almost all of the real numbers. Let alone outside of math.
Of course syntactically you can mix up definition and computation somewhat arbitrarily as you can define “definition”=”computation”=”undefined”=”ojfg99″=”krümpeldefinatorisator”=”????!??!”. That is just a matter of syntax then, and that gets absurdly complex if you produce many complex definitions which might be different than earlier definitions. Hence maybe sometimes the long time it can take to wash all of that away to start with clearer definitions.
Unquestionably if you are not an absolute lover of math it might be a bit boring to talk about definitions, and I am only partially fond of math as bivalent logic and well-defined numbers are only of so much interest to me. That is emojis for the people that want hearts on the other hand: 💚💛🧡❤💜🖤🤎 Also computer are cool haha.🖱👾💻
The theory of classical realizability is a framework for the Curry-Howard correspondence which enables to associate a program with each proof in Zermelo-Fraenkel set theory. But, almost all the applications of mathematics in physics, probability, statistics, etc. use Analysis i.e. the axiom of dependent choice (DC) or even the (full) axiom of choice (AC). It is therefore important to find explicit programs for these axioms. Various solutions have been found for DC, for instance the lambda-term called “bar recursion” or the instruction “quote” of LISP. We present here the first program for AC.
You know you’re arguing with a psychedelic-abusing schizophrenicwho sees things, right? (Haven’t you been online long enough to recognize the very distinctive wordsalad & exclusively verbal style of argumentation, or the free association & redefining in his comments while ignoring widely-accepted physics about spacetime?) Seems like a waste of time, IMO.
I did not know any specifics. I did think it was worth my time start skimming because I have another interesting problem vaguely related; then I thought it was worth my time to understand what the objection buried under the word salad (yes, Benjamin, it’s word salad) might be, because it seemed like there might actually be one. And there was! Standard lambda calculus at face value doesn’t work with nonconstructive proofs. That’s interesting and I didn’t know it. Then:
[Googles]
as expected, looks like there’s plenty of work on this, and there’s nothing actually surprising here. My standard practice after doing something like this is to leave a perfectly-reasonable-if-they-were-reasonable question getting to the heart of what’s up, as I did; I can afford this of course because I’m much less high profile than you, or, y’know, any physicist. :D
Interestingly and a complete aside: I grew up with a close relative who wrote in That Distinctive Style and only later encountered it on the wider internet, and wasn’t that a revelation.
Have you heard of ad hominem? If that is what is called rationality and upvoted on this platform, than there is truly no hope for fruitful conversation on it. Unfortunately this attitude if you continue to uphold is certain to lead to a lot of problems and dissatisfaction for you and others.
What do you think it does to schizo that is not like me and might be genuinely schizophrenic (I did say that I am but technically I got another diagnosis which I think was more accurate) and psychotic and gets verbally attacked with an ad hominem like this? They might start to lash out at others or harm themselves.
Well done, pat yourself on the back.
>distinctive wordsalad
This is about mathematical proof and mathematical definition.
What you are saying is toxic word salad. It is irrational words, which are also literally toxic if engaged in, which I do not, as I keep my distance from such insults.
1=1 is proven to be true.
1=8 is not true, be definition.
That is not word salad, you are being an extremely impolite and contra-rational (as ad-hominem is one of the most irrational fallacies one could think of) and contra-humanistic by creating an ad hominem of a severe diagnosed condition of all things.
I rarely say this, but you and everyone that upvoted this comment should rethink what they are doing, and frankly it would make sense to be ashamed of what you just said.
Feel free to delete your comment if you think it was not appropriate.
It says it enables to associate a program with each proof in Zermelo-Fraenkel set theory. You can also, in principle associate a program to each value of the busy beaver function, or even uncountably many of them (for example for each value of the BB function a complex-valued function that based of the square of the value of the BB function as a constant in each of the complex-valued functions of each value). Even though you cannot compute the values of the busy beaver function.
That means you can associate programs to uncomputable functions, like the function of definition.
I can also associate a program to X = ?; for example rand(60). That however assumes that the natural number in question is a constant, which as I said above does not logically follow from the instruction of choosing one natural number. You can logically instruct something, and do something different. That is not a contradiction in terms of logic.
But as I said above that program would not compute X, it would merely be associated to X, which is a different notion. For example I can associate the BB function to X, but that does not make X computable, and indeed the opposite follows, as X could have the values of the BB function, which is proven to not be computable. So X could be not computable, and as it could be not computable, it is uncomputable in this context, as the question is if you can compute X without first defining it, which you cannot. Once you defined X=3, you can of course program a computer to deduce X=3, but that happens after the definition, and is not the definition of X in this context (which was asked to be done by a person) but a computational replication of an earlier definition, which is still a definition but not a choice, or even the acknowledgement that there is a choice between choice and no choice (meaning just saying X=Y which is a bit of a non-choice in this context but still valid).
You can find programs for axioms, like if have the axiom S(0)=1 I can find a program that display s(0)=1 on my screen. That is just not a definition, it is a replication or association. I can even find a program that displays 0=1 and is correct about that if I define 0 to mean the natural number 0 the symbol 1 to mean the natural number 0. All of these things are possible and valid, although a bit wild syntactically.
However formalism is syntactic, I am talking about semantics here.
You can construct a formal language that way, but you can also not construct it that way.
In terms of bivalent logic, you can make a choice between constructive and non-constructive.
Because you can make the choice for non-constructive math, what I say is still true, as humans can do non-constructive math, as I just displayed to you. The truth of what I say is not dependent on whether someone else uses symbols like I do. It is related to the reality that we can and do reason in non-constructive ways. That is not a choice, so in that semantic sense non-constructive math is not a choice, as others will do it whether you like or not, and you can do it, whether you like or not.
It is true in constructive math you can construct everything. You can construct 1=2 if you define the symbol 2 to mean the natural number S(0). You can do that without a problem, formally speaking. So in terms of symbols, everything can mean everything just purely syntactically speaking.
I can define each finite set of symbols to mean any other finite set of symbols, purely from the rules of syntax.
Indeed you can literally do that, so all symbols mean the same per definition. I guess a computer does that in certain sense when it reads comments as that is just data that is not computed further (usually), meaning all symbols mean ( ).
Semantically what I say is still true. Of course syntactically you can define the set of symbols I said to mean Habeck would be the best Kanzler ever.
And maybe that would also be a good statement semantically speaking, but that is not what the post is about.
I can also change my definitions by the way. So I can define X = 3. I could literally edit the post above, although for the sake of consistency, I would not want to do that. But I define X = 3 now as far as this conversation is concerned, as I could have edited the post above.
If I did that, your “computation” of X := 1 would just be wrong, as X would be defined to be 3. That would be a contradiction as 1 is not equal to 3.😅
So that would show that you cannot compute X, as X has a value of 3. If it has a constant value, you cannot assign it another value, as that leads to a logical contradiction. I hard coded X=3 into the definition, so to speak, and I could do that in the post above, as it is not based on temporal logic but definitional logic. And of course that can be changed in time while still maintaining the order that the post above is the post, and your comment a comment.
It would then make more sense to go the route above to say I use variable [0] and define that. That would also be possible and valid, but it poses the same question, as we merely changed the variable. It does not really matter if you call it [0] or X, except for the purposes of definition.
I am not claiming everything can be defined, not even close to it. In fact math itself can prove there are undefinable real numbers, for example. So a lot of things are undefined. That is just life. Like uncomputational variables, which I use many in my brain, like [omega] which I can uncomputationally define to be anything I want. I could even define it to be an emoji🥨.
But then we are going beyond computation and definition into the realm of images and perception. Interesting.
OK, but the variable name and mapping something to something is not the point here, as that is not a choice of a natural number.
It assumes that you can even computationally determine that there is ANY possible choice of concrete natural number for X.
That is not true. It is logically true that it could be there is not a possible choice of X, and you can choose for example to say it is equal to any of the variables A,B,C,D,AA,AB,AC,AAAAB,ABBBAABA,… or any other number of variables. It could also be meta variables, like meta variable [A] that can be assigned a variable for example AAAAB, or AAAAB. If I assign it the variable AA, [A]=AA, I now know I am talking about AA, which is a natural number of the set of natural numbers.
That did not get me close to actually choosing a natural number and defining X to be that natural number. Physically we could just be creating more variables, without actually defining X to be anything. That is logically perfectly valid, but does not really address the question posed in any way.
There is no computational choice of X, so what you are saying here is a variable. The question is what the variable 1 you are talking about here is. It could mean the pixels on the screen (the symbol 1 as it appears on my or your screen), it could in principle mean a variable symbolized by 1 that can take the value of a natural number.
It could also mean the natural number one, but based on what you said, that would just be my interpretation.
Based on what you said, it is not clear what definition you operate based on.
I am not talking about interpretations here. Well I am, evidently, but here we leave bivalent logic for sure.
There are always many many interpretations for anything. That is another rabbit hole to go down to. Very interesting for sure, but it doesn’t get us any closer to the semantic and epistemological question the post poses. Which is, what is X?
I can just define it to be a natural number 1, but that is not a computation, but a definition, as I define it as such.
[ Epistemic status: way more tentative than it sounds. Talking out my hat. ]
Wait. Definability is beyond computability. Gotcha. But I think there’s a reference problem when you say humans can do “it”. Humans can’t actually solve any of these problems. They can define them. I don’t know how much exploration have been done about whether definitions are computable, and whether a theoretical Turing machine can define problems that it can’t solve.
I suspect the answer is “yes”, definitions can be computable even if the thing defined is not. Which then squares my intuition that humans are actually computing devices, just very complicated ones.
I also object to the sloppy title—“choose one” has a pretty open meaning, and “one” is an absolutely valid choice. It was performed by my brain (or by the section of the universe that computes my thinking, if you prefer), so it was done computationally. Further, obXKCD: https://xkcd.com/221/
>But I think there’s a reference problem when you say humans can do “it”. Humans can’t actually solve any of these problems
You got me here. Problem can be a “trick” term in this regard, as this is a very simple definition. I am just calling it problem to lure people into a computational logic to see for themselves that it makes no sense and is not how they address the question or instruction.🙂
In fact definition is also not necessarily the right term.
Internally I did uncomputationally deduce that X=1, because it is the first natural number and X can be 1, and hence X is 1 in this particular context of me actually writing this text. Of course one could deduce something else, but it is what I deduced.
Only afterwards did I define X=1, as I recognized in principle I could have deduced something else if my natural numbers were sorted differently, but I deduced based on logical order not geometric order.
>I don’t know how much exploration have been done about whether definitions are computable, and whether a theoretical Turing machine can define problems that it can’t solve.
They are not always (in bivalent logic), as that is contradictory (for example I can self-referentially state: “This definition is not computable and X=1”, which is a valid definition, and true by definition, as you can define something to not be computable, even though it is already not computable as a definition is not a computation (for example I could define that the busy beaver function is “really really really not computable”, even though it is already proven it is not computable).
You can define the busy beaver function, but you cannot compute it. The abstract set of Turing machines and their computations define the busy beaver function, but it cannot be computed by a Turing machine. That was proven by contradiction, as you can verify on the internet.
I guess in polyvalent logic you can through all terms together as you have infinite options what the terms can mean. Also a bit of a pitfall, as I think in general polyvalent logic might be be more valid, as it applies in more situations, but in mathematical reasoning bivalent logic is used for good reasons (like if 0s might or might not be 1 your numbers system becomes pretty complex pretty quickly).
>I also object to the sloppy title—“choose one” has a pretty open meaning, and “one” is an absolutely valid choice. It was performed by my brain (or by the section of the universe that computes my thinking, if you prefer), so it was done computationally
Choose one has a well defined meaning. You take one number of the set of all natural numbers, and put it on the right side of the equation. It is open, as the set of natural numbers is infinite.
That is inherent to the process, not sloppy in any way. I would say it is a bit sloppy to suggest otherwise but whatever, I think my term “problem” was also sloppy, so maybe let’s say instruction😉.
You said “one” is an absolutely valid choice, I unfortunately have to say it is literally not a valid choice as it a string not a natural number. It seems you just converted what I said into a string and outputted it back to me. That is a valid computation, it is however not a valid definition for X.
It seems just internally restated your assumption that your brain performs definitions computationally. The question is if that is true?
I would say per definition it is not true as definition and computation have a different semantic meaning, and are also different terms syntactically as definition!=computation. It might be point out the obvious, but although it is obvious we are not always conscious of it. We are not always conscious of any definitions which is maybe because definitions are not that important. Even just in math there is undefinability like the undefinable real numbers which make up almost all of the real numbers. Let alone outside of math.
Of course syntactically you can mix up definition and computation somewhat arbitrarily as you can define “definition”=”computation”=”undefined”=”ojfg99″=”krümpeldefinatorisator”=”????!??!”. That is just a matter of syntax then, and that gets absurdly complex if you produce many complex definitions which might be different than earlier definitions. Hence maybe sometimes the long time it can take to wash all of that away to start with clearer definitions.
Unquestionably if you are not an absolute lover of math it might be a bit boring to talk about definitions, and I am only partially fond of math as bivalent logic and well-defined numbers are only of so much interest to me. That is emojis for the people that want hearts on the other hand: 💚💛🧡❤💜🖤🤎 Also computer are cool haha.🖱👾💻
[Googles] Why does something like https://arxiv.org/pdf/2006.05433.pdf not resolve things? Is it simply wrong? Is it not actually applicable?
You know you’re arguing with a psychedelic-abusing schizophrenic who sees things, right? (Haven’t you been online long enough to recognize the very distinctive wordsalad & exclusively verbal style of argumentation, or the free association & redefining in his comments while ignoring widely-accepted physics about spacetime?) Seems like a waste of time, IMO.
I did not know any specifics. I did think it was worth my time start skimming because I have another interesting problem vaguely related; then I thought it was worth my time to understand what the objection buried under the word salad (yes, Benjamin, it’s word salad) might be, because it seemed like there might actually be one. And there was! Standard lambda calculus at face value doesn’t work with nonconstructive proofs. That’s interesting and I didn’t know it. Then:
as expected, looks like there’s plenty of work on this, and there’s nothing actually surprising here. My standard practice after doing something like this is to leave a perfectly-reasonable-if-they-were-reasonable question getting to the heart of what’s up, as I did; I can afford this of course because I’m much less high profile than you, or, y’know, any physicist. :D
Interestingly and a complete aside: I grew up with a close relative who wrote in That Distinctive Style and only later encountered it on the wider internet, and wasn’t that a revelation.
Have you heard of ad hominem? If that is what is called rationality and upvoted on this platform, than there is truly no hope for fruitful conversation on it. Unfortunately this attitude if you continue to uphold is certain to lead to a lot of problems and dissatisfaction for you and others.
What do you think it does to schizo that is not like me and might be genuinely schizophrenic (I did say that I am but technically I got another diagnosis which I think was more accurate) and psychotic and gets verbally attacked with an ad hominem like this? They might start to lash out at others or harm themselves.
Well done, pat yourself on the back.
>distinctive wordsalad
This is about mathematical proof and mathematical definition.
What you are saying is toxic word salad. It is irrational words, which are also literally toxic if engaged in, which I do not, as I keep my distance from such insults.
1=1 is proven to be true.
1=8 is not true, be definition.
That is not word salad, you are being an extremely impolite and contra-rational (as ad-hominem is one of the most irrational fallacies one could think of) and contra-humanistic by creating an ad hominem of a severe diagnosed condition of all things.
I rarely say this, but you and everyone that upvoted this comment should rethink what they are doing, and frankly it would make sense to be ashamed of what you just said.
Feel free to delete your comment if you think it was not appropriate.
No, I do not think it is wrong.
It says it enables to associate a program with each proof in Zermelo-Fraenkel set theory. You can also, in principle associate a program to each value of the busy beaver function, or even uncountably many of them (for example for each value of the BB function a complex-valued function that based of the square of the value of the BB function as a constant in each of the complex-valued functions of each value). Even though you cannot compute the values of the busy beaver function.
That means you can associate programs to uncomputable functions, like the function of definition.
I can also associate a program to X = ?; for example rand(60). That however assumes that the natural number in question is a constant, which as I said above does not logically follow from the instruction of choosing one natural number. You can logically instruct something, and do something different. That is not a contradiction in terms of logic.
But as I said above that program would not compute X, it would merely be associated to X, which is a different notion. For example I can associate the BB function to X, but that does not make X computable, and indeed the opposite follows, as X could have the values of the BB function, which is proven to not be computable. So X could be not computable, and as it could be not computable, it is uncomputable in this context, as the question is if you can compute X without first defining it, which you cannot. Once you defined X=3, you can of course program a computer to deduce X=3, but that happens after the definition, and is not the definition of X in this context (which was asked to be done by a person) but a computational replication of an earlier definition, which is still a definition but not a choice, or even the acknowledgement that there is a choice between choice and no choice (meaning just saying X=Y which is a bit of a non-choice in this context but still valid).
You can find programs for axioms, like if have the axiom S(0)=1 I can find a program that display s(0)=1 on my screen. That is just not a definition, it is a replication or association. I can even find a program that displays 0=1 and is correct about that if I define 0 to mean the natural number 0 the symbol 1 to mean the natural number 0. All of these things are possible and valid, although a bit wild syntactically.
Depends on your formalism. In constructive type theory, proof are indeed programs. See e.g. https://en.m.wikipedia.org/wiki/Curry–Howard_correspondence
Yes, that is what the link in my post refers to.
However formalism is syntactic, I am talking about semantics here.
You can construct a formal language that way, but you can also not construct it that way.
In terms of bivalent logic, you can make a choice between constructive and non-constructive.
Because you can make the choice for non-constructive math, what I say is still true, as humans can do non-constructive math, as I just displayed to you. The truth of what I say is not dependent on whether someone else uses symbols like I do. It is related to the reality that we can and do reason in non-constructive ways. That is not a choice, so in that semantic sense non-constructive math is not a choice, as others will do it whether you like or not, and you can do it, whether you like or not.
It is true in constructive math you can construct everything. You can construct 1=2 if you define the symbol 2 to mean the natural number S(0). You can do that without a problem, formally speaking. So in terms of symbols, everything can mean everything just purely syntactically speaking.
I can define each finite set of symbols to mean any other finite set of symbols, purely from the rules of syntax.
Indeed you can literally do that, so all symbols mean the same per definition. I guess a computer does that in certain sense when it reads comments as that is just data that is not computed further (usually), meaning all symbols mean ( ).
Semantically what I say is still true. Of course syntactically you can define the set of symbols I said to mean Habeck would be the best Kanzler ever.
And maybe that would also be a good statement semantically speaking, but that is not what the post is about.
I read it—provide a computational witness for ∃X.X∈N.
X:=1 (as in—a partial mapping from variable names to values that maps “X” to 1) is one such witness, and there is no contradiction.
I can also change my definitions by the way. So I can define X = 3. I could literally edit the post above, although for the sake of consistency, I would not want to do that. But I define X = 3 now as far as this conversation is concerned, as I could have edited the post above.
If I did that, your “computation” of X := 1 would just be wrong, as X would be defined to be 3. That would be a contradiction as 1 is not equal to 3.😅
So that would show that you cannot compute X, as X has a value of 3. If it has a constant value, you cannot assign it another value, as that leads to a logical contradiction. I hard coded X=3 into the definition, so to speak, and I could do that in the post above, as it is not based on temporal logic but definitional logic. And of course that can be changed in time while still maintaining the order that the post above is the post, and your comment a comment.
It would then make more sense to go the route above to say I use variable [0] and define that. That would also be possible and valid, but it poses the same question, as we merely changed the variable. It does not really matter if you call it [0] or X, except for the purposes of definition.
I am not claiming everything can be defined, not even close to it. In fact math itself can prove there are undefinable real numbers, for example. So a lot of things are undefined. That is just life. Like uncomputational variables, which I use many in my brain, like [omega] which I can uncomputationally define to be anything I want. I could even define it to be an emoji🥨.
But then we are going beyond computation and definition into the realm of images and perception. Interesting.
OK, but the variable name and mapping something to something is not the point here, as that is not a choice of a natural number.
It assumes that you can even computationally determine that there is ANY possible choice of concrete natural number for X.
That is not true. It is logically true that it could be there is not a possible choice of X, and you can choose for example to say it is equal to any of the variables A,B,C,D,AA,AB,AC,AAAAB,ABBBAABA,… or any other number of variables. It could also be meta variables, like meta variable [A] that can be assigned a variable for example AAAAB, or AAAAB. If I assign it the variable AA, [A]=AA, I now know I am talking about AA, which is a natural number of the set of natural numbers.
That did not get me close to actually choosing a natural number and defining X to be that natural number. Physically we could just be creating more variables, without actually defining X to be anything. That is logically perfectly valid, but does not really address the question posed in any way.
There is no computational choice of X, so what you are saying here is a variable. The question is what the variable 1 you are talking about here is. It could mean the pixels on the screen (the symbol 1 as it appears on my or your screen), it could in principle mean a variable symbolized by 1 that can take the value of a natural number.
It could also mean the natural number one, but based on what you said, that would just be my interpretation.
Based on what you said, it is not clear what definition you operate based on.
I am not talking about interpretations here. Well I am, evidently, but here we leave bivalent logic for sure.
There are always many many interpretations for anything. That is another rabbit hole to go down to. Very interesting for sure, but it doesn’t get us any closer to the semantic and epistemological question the post poses. Which is, what is X?
I can just define it to be a natural number 1, but that is not a computation, but a definition, as I define it as such.