Sage of WisdomSoulofDeity
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- When you say 'tell what a number is', what do you mean by 'is'?
A number is defined by the successor function, so the only relevant information to say what a number 'is' is to the make sure you are able to tell what symbol precedes it, which is no harder then remembering that the decimal string 1010000 is preceded by the decimal string 1009999. You take the lowest branch facing right turning it left (unless it is the top branch in which case it disappears), while turning all branches below it right and you get the symbol that precedes it.
To succeed a number you take the lowest branch facing left and turn it right (or if all values a facing right you add a branch facing right at the top) while turning all branches below it back left.
Its also helpful to be able to multiply and add. The beauty of this notation is that while it works like place value, only small multiplication or addition tables need to be learned. you just need to know
a left branch plus left branch is a left branch
a left branch plus right branch is a right branch
a right branch plus left branch is a right branch
a right branch plus right branch is a left branch carry a right branch over
a left branch multiplied by a left branch is a left branch
a left branch multiplied by a right branch is a left branch
a right branch multiplied by a left branch is a left branch
a right branch multiplied by a right branch is a right branch
compare this to the 10 by 10 multiplication tables that need to be recalled or proven for base ten notation with hindu-arabic numerals.
so arithmetic is easier and counting is just as simple, however there will more place value work to do but no more rules need to be introduced.
some of the symbols I send you last time were defective, here's 0 to 255, the fact that I can produce these symbols so quickly should help you believe that its really easy to tell what values come next. You may notice that these are essentially bytes with bits on or off representing the branches though can be increased to 'by any size'.
once we get really high they'll be a practical (but not geometric) problem with how much detail we can put in the symbol, but that is just like the decimal system where numbers get too long and we prefer to put things in something like standard form or arrow notation, so things like that I may include in the future (as functions of course, since everything is a function in my language)
- sorry I didn't get round to replying, if I hadn't other things to do my language would be more refined and complete.
Arity is present in many functions anyway, as brackets do not always suddenly allow more or less than the required number of input values. Brackets also require more procedures to check whether something is a valid term as the brackets need to be counted and a left bracket can't be found inside a function while it's right bracket is found outside etc . If you like parentheses then its completely fine with me if you explore languages that use them, or develop your own language with them. As well as finding them unnecessary or costly, I also don't like them as I'd like to be able to have a language where every symbol has arity, so I would expect that even if you convinced me that they were necessary or efficient I doubt I'd make my language dependent on them.
That being said the language could be altered very intuitively to include brackets, so don't worry too much. But there are probably even more visually intuitive ways to understand order of operations (like drawing lines underneath the symbol).
As for and unbounded number of characters, regarding characters as geometric objects it is certainly possible (in any computer image you'd eventually run out of pixels, but I see it as the same problem as in place value arabic numbers where you eventually run out of tape). I do so for numbers by stacking branches left or right of a vertical stalk to indicate what it would look like in binary form,
here's an example of natural numbers increasing from 0 to 63
- I hated the fact that they threw Zant out at the last minute, I was really disappointed. He was such an awesome and scary character, you really didn't know what to expect with him around. Ganon was epic, but still. He shoulda been saved for a sequel, tbh.
Yeah there have always been some questionable adult moments in Zelda. And then there's just the regular random weirdos.
- I would have protested your infraction myself! (Oot disliker/MM lover)
Yeah MM is dissed for some pretty stupid stuff. Kinda like Twilight Princess, even though TP deserves some of it. IMO Oot was darker in feel (not theme) than MM. Both should have been rated T for some pretty gnarly schtuff.
Anyways, good post. See you around!
- Well if I wanted to write something that was efficiently translated to a computer I'd use machine code. I'm not trying to write machine code or built the next alternative to the computer there are some very smart people who can do that better than I.
My reader is a specific type of reader its assumed.
They know what all the basic symbols means
They understand juxtaposition in the same way as equational logic was described in PTJ's book.
Given the rules of inference they can deductively close any set of propositions.
They can understand an unbounded number of characters
They can read an unbounded amount of characters.
They can be an unbounded number of states.
This reader may be impossible, but it is an approximation to the ideal tautology searcher who starts with the minimal amount of information.
The assumption here is that the reader will know the symbol for 3 means SSS0 (which is definition of 3), if not then SSS0 will have to be there instead. From that and Peano arithmetic and understanding of arity the reader will be able to determine how to use statements including that value.
- this is not the case, also bear in mind I'm not trying to write a computer language, just something that resembles a formal language.
lets let M and P be the multiply and add functions each with arity 2.
every natural number can be expressed as a single character (as opposed to the arabic notation 1,2,3,4,... you are using where you need to start using multiple characters to denote larger numbers e.g. 145 is three characters long), so characters x, y and z represents numbers. And we can form the statement
here we've embedded operations without parentheses.
Assuming you have a character for each x,y,z you can't do better than that, without using the rules of multiplication/addition to simplify. Also I deal with algebra on a general level with non associate operations so they'll be operators, say G (of arity 2) where GaGbc does not equal GGabc.
I'll try and work a base system in, but it'll end up being less efficient then typical mathematical language. I'm thinking along the lines of
[Base Function Symbol][Base Value(e.g. 2 or 10)][Number of digits][digit1][digit 2]....etc.
It needs 3 more symbols than usual numbers however my language is not designed around standard base 10 notation, for better or for worse it's designed around juxtaposition meaning 'next input value' universally, as well there being a single symbol for each natural number.
I'll could develop symbols to denote derived operations 'both', 'either' and 'not' and I have a symbol for 'equality' but they aren't necessary, if I treated these as basic operations I'd need more than my avatar's size to model propositional calculus, but apart from being useful in the real world, we don't need to bias these particular ones, we have a way of talking about anything boolean using 'false' and 'if', why should we be biased towards 'p and q' as opposed to something like TTpTqTTrsTuvw, other than for human use and understanding? And is useful, but it isn't any more meaningful.
but any writer can specify a symbol (from a choice of countably infinite symbols) to mean 'both', 'either', or 'not' before they write everything else. In fact they could specify a symbol say D so that we can use Dpqrsuvw to mean TTpTqTTrsTuvw if they wanted to. I'm know other languages allow you to create derived operations.
Not to say my language doesn't retain a lot of bias towards human understanding, for example it's visible, written on tape, and I've introduced unnecessary characters just to make things easy (in fact in order to say all natural numbers you only need two things; a constant 0 and 'S' a single arity successor function. For example three would be SSS0, but this gets long fast and so I wrote in an alternative), but the aim here is to explore alternatives to 'everyday' biased approaches to languages.
- A2 putting aside my symbols. I find that the notation you are using has an efficiency problem with necessary parentheses (brackets), when I wrote these out as an explanation I changed it back to a more conventional notation since that would be easier to understand, but I'm a fan of no parentheses, despite having to use them all the time.
I can write that without parentheses by defining Txy to mean (x?y), because T ( it looks like the symbol I chose)has an arity of 2 we know to end after two terms.
so '(p?(q?p))' would become 'TpTqp'. And this works because for the first T the input is the terms 'p' and 'Tqp', 'Tqp' being a valid term because it has input of two variables 'q' and 'p'. Notice how we need fewer characters to write this now and also that if you removed the brackets in that C notation you get p?q?p which can be one of two different things (one is a tautology the other isn't)
There's a simple algorithm in my notation for checking if something is a valid term, start at the place after the right hand end of the string with the counter set to 0. Moving left, increase by 1 for each variable and decrease by n-1 for each function of arity n-1, the term is valid iff the counter never falls below 1 after starting and it ends on 1.
so the 5th line with the counter underneath
It is a definition of if and then statements, but since we have a 'false' proposition (here it looks like =)(which also means contradiction) can be extended to other boolean statements.
we can define the statement 'not x' as 'if x, contradiction'. 'Tx='
we can define the statement 'either x, y' as 'if not x, y', so in full that would be 'TTx=y'.
we can define the statement 'both x,y' as 'not either not x, not y' in full is 'TTTTx==Ty=='
here I have variables and constants in terms of propositions, for the sake of arity () I've needed to include natural numbers though (which are constants but not propositions). There is one novelty about my language and that is that I have countably infinitely many symbols for the natural numbers, but I aim to have functions that can represent things in base 10 (or other bases) for convenience.
At the moment I'm in the realm of propositional calculus which means I can talk about boolean statements on proposition, but my language has no quantifiers and 'equality' is still a difficult operation.
My next move will be to extend my language so that a can talk about predicate calculus, where I can begin writing statements (peano arithmetic Peano axioms - Wikipedia, the free encyclopedia) on how these natural numbers interact under newly defined operations (which will be addition and multiplication). But I need to get through predicate calculus first which is fiddly.
Of course when I get round to addition and multiplication the operator will be consistently placed on the left of the term.
I got most inspiration for doing this from my lecturer's book Amazon.com: Notes on Logic and Set Theory (Cambridge Mathematical Textbooks) (9780521336925): P. T. Johnstone: Books, but there are probably better books on formal languages for non mathematicians.
- lol it wasn't meant to look like hylian, but that's a plus.
Its a prototype for a new mathematical language I'm writing. The statements here are axiomatising propositional calculus which is a very simple form of logic.
The explanation as to how it works is quite long...
so from top left to bottom right the symbols are as follows
'equality function', 'arity function', 'false function', '0', 'implication function', '2', 'modus ponens relation', '3'... and the rest are just algebraic symbols to denote propositions.
the arity function works as follows, when you place your arity function symbol you place your function to the right and then this symbolises the number of input variable the function needs.
the equals function acts as follows, you place the equals symbol, then you place to the right your first value (say a) then you place your second value (say b) next to that on the right. Then you have written the equality statement between these two values (a=b).
when writing with other functions, when we want to write the function's value when a load of values has been inputted. we start on the left with the function symbol, then to the right we write all the values (denoted by symbols or strings representing functions of symbols, as this works recursively). There should be the arity number of input values and when we run over the arity number of input values we stop considering characters on the right as input variables.
so as an important example my avatar means this
we have a constant proposition to denote false (a false function that acts on 0 propositions)
we act the implication function on 2 propositions
we act the modus ponens truth function on 3 propositions
if p then (if q then p)
if (if x then (if y then z)) then (if (if x then y) then (if x then z))
if (if (if w then false) then false) then w
we can obtain t via modus ponens from (s) and (if s then t)
the beauty is that from all of this we get can obtain via modus ponens every tautology we need in propositional calculus. We can say this in English, but I don't find English to be a very mathematical language.
And it's its beauty as to why I would choose it as an avatar.
My explanation was probably inadequate, but I'll answer any questions on it if you even have any.
- 1. Aight
2. ...What I was saying is that the reason those leftover items where there was because it was just built right over on Wind Wakers engine. It always was planned to have the more realistic look.
3. No, the IGN video came after. And the reason it looks similar is because it was partly based off Twilight Princess.
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