[HN Gopher] One Plus One Equals Two (2006)
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One Plus One Equals Two (2006)
Author : lemper
Score : 85 points
Date : 2024-10-22 08:21 UTC (14 hours ago)
(HTM) web link (blog.plover.com)
(TXT) w3m dump (blog.plover.com)
| dvh wrote:
| 1+1=3 (for very large values of 1)
| dist-epoch wrote:
| For extreme values 1+1 can be as high as 5.
| marcosdumay wrote:
| It's between 0 and 10, and can be approximated by either
| depending on the context...
| croes wrote:
| And 1x1=2 according to Terrence Howard
| omeysalvi wrote:
| Actually, it is a metaphor for formulating a brand new branch
| of mathematics that fixes the identity principle and all the
| problems with the square root of two. But also, it is not a
| metaphor because show me any physical system where an action
| times an action does not equal a reaction.
| feoren wrote:
| It's actually super easy to form a "brand new branch of
| mathematics". Just start with some definitions and run with
| them. Although you'll almost certainly end up with
| something inconsistent. And if you don't, it'll almost
| certainly be not useful. And if it is useful, it'll almost
| certainly turn out to be the exact same math just wearing a
| costume.
|
| There are no problems with the square root of two.
|
| > show me any physical system where an action times an
| action does not equal a reaction.
|
| Show me any gazzbok where a thrushbloom minus a grimblegork
| does not equal a fistelblush. Haha, you can't do it, can
| you!? I WIN!
|
| That is to say: you're using silly made up definitions of
| "action" and "times" here.
| ndsipa_pomu wrote:
| > show me any physical system where an action times an
| action does not equal a reaction
|
| Not quite sure what an action times an action is, but how
| about rotating a 2d shape 180 degrees? Do that twice and
| it's the same as not rotating it at all.
| croes wrote:
| You mean two reactions. Otherwise 1x1 would be 1
| Suppafly wrote:
| Are you saying you actually buy into the Terrence Howard
| school of mathematics? For serious?
| bluGill wrote:
| I know of 7 different ways to do 1+1 getting 5 different
| answers. I use most of them in my day to day work as a
| programmer. Most of the time 1+1=10 because as a programmer I
| work in binary.
| yjftsjthsd-h wrote:
| > Most of the time 1+1=10 because as a programmer I work in
| binary.
|
| Really low level embedded work? Most programming I know about
| effectively works in base 10 or sometimes hex.
| bluGill wrote:
| Embedded work - not very low level, but I need to decode a
| lot of CAN network packets where the individual bits
| matter. Most of them time I use a hex representation, but
| that is because hex makes it really easy to figure out the
| binary going on underneath. Even when I'm doing normal math
| though it is important to remember that it is binary under
| it all and so overflow happens at numbers that make sense
| in binary terms.
| nwnwhwje wrote:
| 1+1=10 if math were invented before fingers.
|
| Also:
|
| 1 + 5 = 6
| earthboundkid wrote:
| Yi + Yi = ni.
| somat wrote:
| I would say 1 + 1 = 4 for very large values of one.
|
| You only need mid values of 1 for 1 + 1 to equal 3
| youoy wrote:
| Thanks for sharing! I like to look at this example inside the
| debate of if mathematics are invented or discovered.
|
| > That is how Whitehead and Russell did it in 1910. How would we
| do it today? A relation between S and T is defined as a subset of
| S x T and is therefore a set.
|
| > A huge amount of other machinery goes away in 2006, because of
| the unification of relations and sets.
|
| Relations are a very intuitive thing that I think most people
| would agree that are not the invention of one person. But the
| language to describe them and manipulate them mathematically is
| an invention that can have a dramatic effect on the way they are
| communicated.
| benlivengood wrote:
| I'd say mathematics is discovered and definitions are invented.
| E.g. "ordered pair" is not part of set theory, it's an invented
| name we give to a convenient definition of a set schema.
|
| Even base-N representations are an invention: S() and zero are
| all you need, but Roman Numerals were an improvement over
| base-1 representations and base-N is significantly more
| convenient to work with.
| nyrikki wrote:
| Be careful with making assumptions from modern, formalized
| set theory and the naive set theory.
|
| The axiom schema of specification is added to avoid Russell's
| paradox.
|
| A set in the naive meaning is just a well-defined collection
| of objects.
|
| As ordered pairs are a binary relation, foundedness or order
| are operation dependant, and assuming an individual set is
| unordered is a useful assumption.
|
| But IMHO it is problematic from a constructivist mathematics
| perspective. The ambiguity of a nieve set, especially when
| constricting the natural numbers, which are obvious totally
| ordered is a challenge to overcome.
|
| I know the Principia was focused on successor sets, so mostly
| avoid it, but IMHO they would have hit it when trying to
| define an equally operation
|
| If you remember membership and not elements define a set:
|
| {a,b,c}=={a,b,c,b}=={c,b,b,a}
|
| In a computing context, there were some protocols that may
| have been IBM specific that required duplicate members to be
| adjacent.
|
| So while the first and the third sets would be equivalent,
| the second wouldn't be, so order mattered.
|
| Most actual implementations just dropped the redundant
| elements, vs track membership, but I was just trying to
| provide an actual concrete example.
|
| IIRC the axiom schema of specification is one of those that
| was folded into others in modern ZFC textbooks so it is easy
| to miss.
| benlivengood wrote:
| I'm not sure if I completely understand your point. Is it
| that the definitions of ordered pairs must be done
| carefully when talking about constructions in Principia
| because of its formulation in logical predicates, e.g. care
| was taken when constructing sets to avoid Russell's paradox
| explicitly given the axioms of logic rather than Russell's
| paradox being excluded in ZF by the axiom schema of
| specification?
|
| Or is the difficulty in introducing a canonical order for
| the ordered pair, or introducing well/partial-ordering in
| sets themselves? I guess I see an ordered pair as more of
| an indexical definition than an ordering definition.
| kevin_thibedeau wrote:
| Mathematics is entirely founded on human invention.
| benlivengood wrote:
| When we wrote simple mathematics on the Pioneer and Voyager
| probes I think it was under the assumption that anyone or
| anything else finding them would have co-discovered enough
| mathematics to recognize it on the plaques. That's the
| sense in which I use the word "discovered" for much of
| mathematics. Our definitions will differ from aliens but
| the foundations will be translatable.
| kevin_thibedeau wrote:
| A sentient entity could well decide to simulate the
| universe without developing tools to approximate it.
| yohannparis wrote:
| Thank you, it's an interesting read, because on my own, without
| the explanation this will have been over my head.
| pvg wrote:
| The mentioned size and density of Whitehead & Russel's
| _Principia_ make the few dozen pages of Goedel 's _On Formally
| Undecidable Propositions of Principia Mathematica and Related
| Systems_ one of the greatest "i ain't reading all that/i'm happy
| for u tho/or sorry that happened" mathematical shitposts of all
| time.
| oglop wrote:
| Godel had great respect for their work and was considered one
| of only a few people at the time to have read and understood
| the work. He wrote an entire paper later in life explaining he
| wouldn't have come to his result without Principia because it
| showed him a base case to work from. Him and Russell would
| continue to meet and discuss logic well into the 50's.
| awanderingmind wrote:
| That was a lovely read, thank you. I particularly enjoyed the
| analogy between 'a poorly-written computer program' (i.e. one
| with a lot of duplication due to inadequate abstraction), and the
| importance of using the appropriate mathematical machinery to
| reduce the complexity/length of a proof. It brings the the Curry-
| Howard isomorphism to mind:
| https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...
| cubefox wrote:
| Oh, so the l in lambda calculus was just a poor man's circumflex.
|
| Unrelated, but why doesn't Hacker News have support for latex?
| And markdown, for that matter?
| gabrielsroka wrote:
| It supports https://news.ycombinator.com/formatdoc
| yjftsjthsd-h wrote:
| Sure, but that's it's own not-quite-markdown thing, which is
| extra annoying because it's _just_ close enough that people
| think it is markdown and do things like writing code blocks
| with ```. IMHO it 'd be much better to just actually do
| markdown, or at least a strict subset.
| oersted wrote:
| One of the best things about Markdown is that it is also a
| great plain text format for when rendering is not
| available.
|
| But I do agree that HN's format should be a strict subset,
| it is so close.
| wholinator2 wrote:
| Yeah but could it even be changed at this point? I'd
| imagine that once the ball gets rolling, changing any kind
| of formatting rules for a site with over a decades worth of
| (hundreds of thousands, tens of millions? ) of posts would
| be pretttty hard to get past committee
| yjftsjthsd-h wrote:
| I would strongly favor writing a script that went through
| the database and rewrites existing comments from the old
| to new syntax; I believe in this case that's doable. And
| you would want to message it ahead of time of course. But
| with those things done I think it'd work fine, especially
| because I suspect virtually anyone who's gotten used to
| the HN formatting codes is already familiar with real
| markdown so it'd be a relatively painless transition.
| cubefox wrote:
| Simple solution: apply the new formatting code only to
| new comments, that is, comments written after the date
| the new formatting was supported.
| cubefox wrote:
| That's only very limited support of the most basic forms of
| formatting. It's the year 2024, and Hacker News can't do
| better? Even the blog post above, from 2006, uses a LaTeX
| plugin.
| redbell wrote:
| I often use the analogy "1+1=?" in debates with both friends and
| strangers, especially when discussing subjective topics like
| politics, religion, and geopolitical conflicts. It's a simple way
| to highlight how different perspectives can lead to vastly
| different conclusions.
|
| For instance, I frequently use the example "1+1=10" in binary to
| illustrate that, while our reasoning may seem fundamentally
| different, it's simply because we're starting from different
| premises, using distinct methods, and approaching the same
| problem from unique angles.
| feoren wrote:
| 1 + 1 = Two.
|
| One plus one equals two.
|
| One + 0x01 [?] 2.0
|
| 1+1=10 (in binary)
|
| None of these are "vastly different conclusions". None of these
| are starting from different premises. None of these are using
| different reasoning. You're literally just writing it
| differently. Okay, so? This is a pointless distinction that
| doesn't even apply in a verbal debate at all. It'd be like
| having a philosophical debate with someone and them suddenly
| saying "oh yeah, but what if we were arguing _in Spanish!?_
| Wouldn 't that BLOW YOUR MIND!?" No? It has absolutely nothing
| to do with anything. I would be annoyed at you if you tried to
| use this in an argument with me.
| hks0 wrote:
| > It's a simple way to highlight how different perspectives can
| lead to vastly different conclusions.
|
| But 1+1=10 and 1+1=2 are not different conclusions, they are
| precisely the same conclusions but with different
| representations.
|
| A better example might be 9 vs 6 written on the parking floor:
| depending on where you're standing, you'll read the number
| differently (and yet one of the readings is wrong).
| tmtvl wrote:
| > _(and yet one of the readings is wrong)._
|
| It may not even be a number which is written, but the
| hiragana no ( _no_ ).
| earthboundkid wrote:
| It could be Japanese beeper slang and mean Q.
| Tainnor wrote:
| > theorems like *22.92: a[?]b-a[?](b-a)
|
| Either I misunderstand the notation or there seems to be
| something missing there - the right hand side of that implication
| arrow is not a formula.
|
| I would assume that what is meant is a[?]b-a[?](b-a)=b
| adrian_b wrote:
| The main point of the parent article is not about 1+1=2, but
| about the importance of the concept of ordered pair in
| mathematics and about how the introduction and use of this
| concept has simplified the demonstrations that were much too
| complicated before this.
|
| While the article is nice, I believe that the tradition
| entrenched in mathematics of taking sets as a primitive concept
| and then defining ordered pairs using sets is wrong. In my
| opinion, the right presentation of mathematics must start with
| ordered pairs as the primitive concept and then derive sequences,
| sets and multisets from ordered pairs.
|
| The reason why I believe this is that there are many equivalent
| ways of organizing mathematics, which differ in which concepts
| are taken as primitive and in which propositions are taken as
| axioms, while the other concepts are defined based on the
| primitives and other propositions are demonstrated as theorems,
| but most of these possible organizations cannot correspond to an
| implementation in a physical device, like a computer.
|
| The reason is that among the various concepts that can be chosen
| as primitive in a mathematical theory, some are in fact more
| simple and some are more complex and in a physical realization
| the simple have a direct hardware correspondent and the complex
| can be easily built from the simple, while the complex cannot be
| implemented directly but only as structures built from simpler
| components. So in the hardware of a physical device there are
| much more severe constraints for choosing the primitive things
| than in a mathematical theory that only describes the abstract
| properties of operations like set union, without worrying how
| such an operation can actually be executed in real life.
|
| The ordered pair has a direct hardware implementation and it
| corresponds with the CONS cell of LISP. In a mathematical theory
| where the ordered pair is taken as primitive and sets are among
| the things defined using ordered pairs, many demonstrations
| correspond to how various LISP functions would be implemented.
| Unlike ordered pairs, sets do not have any direct hardware
| implementation. In any physical device, including in the human
| mind, sets are implemented as equivalence classes of sequences,
| while sequences are implemented based on ordered pairs.
|
| The non-enumerable sets are not defined as equivalence classes of
| sequences and they cannot be implemented as such in a physical
| device but at most as something of the kind "I recognize it when
| I see it", e.g. by a membership predicate.
|
| However infinite sets need extra axioms in any kind of theory and
| a theory of finite sets defined constructively from ordered pairs
| can be extended to infinite sets with appropriate additional
| axioms.
| wildermuthn wrote:
| "1 + 1 = 2" is only true in our imagination, according to logical
| deterministic rules we've created. But reality is, at its most
| fundamental level, probabilistic rather than deterministic.
|
| Luckily, our imaginary reality of precision is close enough to
| the true reality of probability that it enables us to build
| things like computer chips (i.e., all of modern civilization).
| And yet, the nature of physics requires error correction for
| those chips. This problem becomes more obvious when working at
| the quantum scale, where quantum error correction remains
| basically unsolved.
|
| I'm just reframing the problem of finding a grand unified theory
| of physics that encompasses a seemingly deterministic macro with
| a seemingly probabilistic micro. I say seemingly, because it
| seems that macro-mysteries like dark matter will have a more
| elegant and predictive solution once we understand how micro-
| probabilities create macro-effects. I suspect that the answer
| will be that one plus one is _usually_ equal to two, but that
| under odd circumstances, are not. That's the kind of math that
| will unlock new frontiers for hacking the nature of our reality.
| earthboundkid wrote:
| Wait, am I crazy for thinking relations are not sets? Two sets
| can be coextensive without the relation have the same intension,
| no? Like the set of all Kings of Mars and the set of Queens of
| Jupiter are coextensive, but the relations are different because
| they have different truth conditions. Or am I misunderstanding?
| JadeNB wrote:
| > Wait, am I crazy for thinking relations are not sets? Two
| sets can be coextensive without the relation have the same
| intension, no? Like the set of all Kings of Mars and the set of
| Queens of Jupiter are coextensive, but the relations are
| different because they have different truth conditions. Or am I
| misunderstanding?
|
| No-one can stop you from using terms as you please and
| investigating their consequences, but, at least in modern
| mathematical parlance, a binary relation is the set of ordered
| pairs that are "related" by it. (Your relation would seem to be
| just a bare set, or perhaps a unary relation, not a binary
| relation which I think is what is often meant without default
| modifier.)
| Animats wrote:
| It's easier if you start from something closer to Peano
| arithmetic or Boyer-Moore theory. I used to do a lot with
| constructive Boyer-Moore theory and their theorem prover. It
| starts with (ZERO)
|
| and numbers are (ADD1 (ZERO)) (ADD1
| (ADD1 (ZERO)))
|
| etc. The prover really worked that way internally, as I found out
| when I input a theorem with numbers such as 65536 in it. I was
| working on proving some things about 16-bit machine arithmetic,
| and those big numbers pushed SRI International's DECSystem 2060
| into thrashing.
|
| Here's the prover building up basic number theory, one theorem at
| a time.[1] This took about 45 minutes in 1981 and takes under a
| second now.
|
| Constructive set theory without the usual set axioms is messy,
| though. The problem is equality. Informally, two sets are equal
| if they contain the same elements. But in a strict constructive
| representation, the representations have to be equal, and
| representations have order. So sets have to be stored sorted,
| which means much fiddly detail around maintaining a valid
| representation.
|
| What we needed, but didn't have back then, was a concept of
| "objects". That is, two objects can be considered equal if they
| cannot be distinguished via their exported functions. I was
| groping around in that area back then, and had an ill-conceived
| idea of "forgetting", where, after you created an object and
| proved theorems about it, you "forgot" its private functions.
| Boyer and Moore didn't like that idea, and I didn't pursue it
| further.
|
| Fun times.
|
| [1] https://github.com/John-
| Nagle/pasv/blob/master/src/work/temp...
| anthk wrote:
| The Computational Beauty of Nature has a tiny Lisp implementing
| integers and aritmethics by hand too, by consing t's.
| anthk wrote:
| The Computational Beauty of Nature shows that with Lisp.
| jk4930 wrote:
| In the same spirit, why 2 + 2 = 4.
|
| https://us.metamath.org/mpeuni/mmset.html#trivia
|
| https://us.metamath.org/mpeuni/2p2e4.html
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