2 + 2 = 4

[Yazata]

This thread is a spinoff from one in Site Feedback, moved to what seems like a more appropriate place.

Regarding 2 + 2 =/= 4, I have two comments

1. Consider modular arithmetic, performed on the number line 1,2,3,1,2,3,1,2,3...

2 + 2 = 1

Where 1 is congruent with 4

Modular arithmetic - Wikipedia

en.wikipedia.org

2. More fundamentally, how is mathematical necessity (or any kind of necessity) discerned?

It seems to me to reduce to little more than intuition, our sense of 'how could it be any other way?'

The skeptic in me makes me a little doubtful about arguments based on intuition. Unfortunately, all of human reason seems to me to reduce to intuition.

nsNS

 

[DaveC426913]

This thread is a spinoff from one in Site Feedback, moved to what seems like a more appropriate place.

Regarding 2 + 2 =/= 4, I have two comments

1. Consider modular arithmetic, performed on the number line 1,2,3,1,2,3,1,2,3...

2 + 2 = 1

Where 1 is congruent with 4

Well ... it isn't really congruent with 4.

Modulo is a function. It takes an input (in this case, 4) and produces an output (1). That's not a 1:1 correlation between input and ouput.

There is no reversing the 1 to get the 4 back, since the 1 could easily have been produced from an infinite number of inputs, such as 7 or 10, 13, etc.

It seems to me to reduce to little more than intuition, our sense of 'how could it be any other way?'

That's what axioms are for.
Our most common mathematical foundations start with the axiom that 0+1=1. That cannot be proven rigorously from first principles.
We must state it as an axiom (although by convention it almost always goes without saying) upon whoch the rest of our math is then built.

 

[axocanth]

2. More fundamentally, how is mathematical necessity (or any kind of necessity) discerned?

It seems to me to reduce to little more than intuition, our sense of 'how could it be any other way?'

The skeptic in me makes me a little doubtful about arguments based on intuition. Unfortunately, all of human reason seems to me to reduce to intuition.

nsNS

I suppose like almost everything else in philosophy the analyticity and necessity of mathematical statements can be -- and has been -- challenged.

I'd also hasten to point out, though, that one does not prove that "2 + 2 = 4" is not necessarily true by saying the very same thing another way -- by switching into Swahili, for instance, or Base Three.

As I'm sure you know, J. S. Mill believed the statements of mathematics were empirical - derived from observation, thus presumably as fallible as any other observation statement. As you doubtless also know, Quine denies the analytic-synthetic distinction altogether: any statement can be held true "come what may" and, conversely, all statements are subject to revision. The reason we tend not to revise the statements of logic and mathematics, on Quine's view, is a "principle of minimal mutilation" -- to revise something like "2 + 2 = 4", though possible in principle, would have such enormous ramifications for our entire holistic "web of belief" that it's far more likely we'd make adjustments elsewhere to maintain overall coherence. (But see quantum logic !).

I think you're right about "intuition", though. The arguments I've seen for the necessity of certain statements, including those of mathematics, take the form of conceivability arguments, exemplified in "possible worlds semantics". The reason we believe it is necessarily true that "2 + 2 = 4" or that "water = H2O" is that it is literally inconceivable to us -- it defies our intuitions -- that it could be any other way.

At this point, other members might retort "Rubbish! I can conceive of water having a chemical formula other than H2O." Putnam et al will gently guide you to see that you're conceiving incorrectly - think about it more carefully and you'll come to see that what you're imagining is not water. But what if we're the ones who are not "conceiving correcting"?

Kripke highlights the fragility of these conceivability arguments with his famous "modal argument" against the reduction of mental states (e.g. pain) to physical brain states (e.g. C-fiber firings). The reduction fails, Kripke argues, because if it were true that pain is identical to some physical brain state it would have to be necessarily identical to that brain state. But there is no necessary identity, Kripke goes on, we can conceive of pain without C-fibers firing in the brain. QED?

Well, he seems to have convinced most philosophers; not necessarily [sic] all.

Putnam gets up to similar mischief on "Twin Earth", of course, also basing his arguments on conceivability and intuition. Then again, linguists do the same thing, testing their theories of generative grammar against what we intuitively feel are grammatically well formed sentences.

If you're not careful you're liable to be led up the garden path. Burp!

Any thoughts?

 

[Ken Fabian]

Not a mathematician, but seems reminiscent of arguing internal angles of triangles don't always add to 180 by positing a curved surface, not a plane, ie allowing curved lines in place of straight ones, ie redefining the context. Sounds like cheating.

Modular arithmetic may legitimately posit a recurring (circular) sequence without beginning or end - 1,2,3, 1,2,3 where adding 2 to 2 gets 1 - and where it is different to numbers with different bases by not (in the clock analogy) counting how many turns? It may be valid in that context but to me it doesn't look anything like a proof that 2+2 does not always equal 4.

 

[Gawdzilla Sama]

Spherical geometry is a unique case, so it's not cheating, its... cheating within the rules.

 

[DaveC426913]

Not a mathematician, but seems reminiscent of arguing internal angles of triangles don't always add to 180 by positing a curved surface, not a plane, ie allowing curved lines in place of straight ones, ie redefining the context. Sounds like cheating.

Cheating?

Imagine if someone said "There is no circumstance in which the sum of the angles of a triangle do not add up to 180 degrees."
Is that a true statement?

The phrase "...there is no circumstance in which..." is tantamount to saying "...there exists no context in which..."

 

[DaveC426913]

Spherical geometry is a unique case,

It's case that falls within "all" circumstances.

so it's not cheating, its... cheating within the rules.

It's not cheating in any fashion.

The statement that "there is no circumstance" attempts to address all circumstances, of any sort, anywhere.

Its pretty easy to find an exception to a statement that tries to claim universality.

 

[axocanth]

Imagine if someone said "There is no circumstance in which the sum of the angles of a triangle do not add up to 180 degrees."
Is that a true statement?

The phrase "...there is no circumstance in which..." is tantamount to saying "...there exists no context in which..."

And imagine if someone tried to refute it by saying:

" "The sum of the angles of a triangle adds up to 一百八十 degrees" is a true statement"


Non-Chinese speakers are left to guess what " 一百八十 " means in Chinese. (Clue: highest possible score with three darts)

If not, translate "180" into Base 3.

 

[exchemist]

Cheating?

Imagine if someone said "There is no circumstance in which the sum of the angles of a triangle do not add up to 180 degrees."
Is that a true statement?

The phrase "...there is no circumstance in which..." is tantamount to saying "...there exists no context in which..."

And of course the angles of a sufficiently large triangle in space do not in general add up to 180 degrees, if GR is right. Or even on the surface of the Earth, for that matter.

 

[Pinball1970]

And of course the angles of a sufficiently large triangle in space do not in general add up to 180 degrees, if GR is right. Or even on the surface of the Earth, for that matter.

Yep, 180 degrees for the sum of a triangle goes out of the window in non Euclidean geometry.

 

[Gawdzilla Sama]

It's not cheating in any fashion.

Ah, an absolute. Love 'em.

 

[axocanth]

Yep, 180 degrees for the sum of a triangle goes out of the window in non Euclidean geometry.

Which raises the interesting question . . . Were our forebears wrong in asserting that the angles of a triangle necessarily add up to 180 degrees? Did we discover that not all triangles have angles that add up to 180 degrees?

Or has there been a change in meaning of the word triangle? (Thus our forebears were not wrong at all).

Similarly, did we discover that the whale is not a fish? Were the ancients wrong to assert that the whale is a fish? Or, by adjusting or refining the meaning of certain terms, have we made it the case that the whale is not a fish?

For more on this, and other exciting developments, try this absolute gem of a book by Joseph LaPorte:

https://www.amazon.com/Natural-Conceptual-Cambridge-Studies-Philosophy/dp/0521825997/ref=sr_1_2?crid=342TDHFT5QCO5&dib=eyJ2IjoiMSJ9.FT4YVxQ_KOzDUnWROg1OqtXNVNa5aNjbqnu96X3tp0orxMOpKaN_FxtnfTgTp7RxtSJ-huhXTBIoaM4qpDk8a5mKhmAD9AZ0b7HyEwwVxLZnXN1wZRc6V3ufg9yckwlaNp5wjgeHwrNuTztDK0epsQtfpgGNSOYUJPbpBM-f5y0bNaV6IXeA450xApcOmDB7M-tKa8kgVedUWXTJaaTNYgfHHJVeCrFqgR3ZwwNO2Cc.ziKipuIvk_tFubxiGHCb_EushFKS9gxV_Itqw5IuYes&dib_tag=se&keywords=Joseph+Laporte&qid=1728254704&s=books&sprefix=joseph+laporte%2Cstripbooks-intl-ship%2C464&sr=1-2

"According to the received tradition, the language used to refer to natural kinds in scientific discourse remains stable even as theories about these kinds are refined. Hence, scientists discover, rather than stipulate, that sentences like 'Whales are mammals, not fish' are true. In this illuminating book, Joseph LaPorte argues that scientists do not discover that sentences about natural kinds, like 'Whales are mammals, not fish', are true rather than false. Instead, scientists find that these sentences were vague in the language of earlier speakers, and they refine the meanings of the relevant natural-kind terms to make sentences true. Hence, scientists change the meanings of these terms. This conclusion prompts LaPorte to examine the consequences of this change in meaning for the issue of incommensurability and for the progress of science."

 

[Ken Fabian]

Isn't the primary definition of a triangle innately "Euclidian"? Straight lines connecting 3 points is a triangle - which are also points on a flat plane. Seems to me positing a not-flat plane and calling the geodesic lines connecting 3 points on it a triangle despite not being straight lines sounds like cheating. It is a shifting of the goalposts at the least, via differences to what the terms mean, (straight, flat) perhaps a consequence of failing to come up with different names for a group of different geometric shapes sharing similarities. As with 2+2 not always equaling 4 it maybe being a way to introduce concepts like modular arithmetic to students, "not all triangles" is a way to introduce non-Euclidian frames of reference. As proof that triangles don't always have internal angles adding to 180, not so much.

 

[axocanth]

Jinx!!!!!!!!!!!!!!!!!!!!!

 

[DaveC426913]

Isn't the primary definition of a triangle innately "Euclidian"?

I do not think so, no.

Straight lines connecting 3 points is a triangle - which are also points on a flat plane.

You might have to delve into the definition of a triangle in the context of non-Euclidean spaces. I'm not prepared to state it, but I'm pretty sure triangles do just fine in non-Euclidean spaces.

As with 2+2 not always equaling 4 it maybe being a way to

And, in fact, we are drifting away from the topic at-hand, stated as-is. It is a fine topic for discussion - but triangles as an analogy gets a little too far from 2+2=/=4.

 

[Gawdzilla Sama]

Straining at gnats and swallowing camels. Nothing new.

 

[C C]

[...] 2. More fundamentally, how is mathematical necessity (or any kind of necessity) discerned?

It seems to me to reduce to little more than intuition, our sense of 'how could it be any other way?'

The skeptic in me makes me a little doubtful about arguments based on intuition. Unfortunately, all of human reason seems to me to reduce to intuition.

If a formal system has a bottom level of properties, rule-abiding relationships, and operations (actions) that everything else has to conform to or fall out of... Then that's arguably where everything becomes brute or resistant to further analysis. "It just beez that way."

With mathematics though... The basic attribute of quantity was historically abstracted from groups of objects that were empirical slash concrete: A measurement of apples or fish in a basket, etc. So the origins of "5" or "6" or units of quantity (and doing things with those symbols) ultimately trail into what seems to be "necessary" features of everyday reality or space and time as consciously encountered. I mean, one can pretend that it fully floats on its own (like old-time mathematical purism), but the primeval beginnings conflict with that idealism.

That said, though, I suppose a symbol manipulation game could be created solely from scratch. One that employs -- in contrast to quantity -- wholly imaginary or non-empirical properties.

The key is consistency -- maintaining the identities assigned to the game's tokens throughout any of its processes. Again: The properties, lawful relationships, and actions that they represent.

But again -- it would surely require multiple strata. The "characteristics" (composite signs or whatever) of the "interesting" level(s) arising from rule-like connections of a primary stratum of signs.

Yet it would require order to produce the symbol manipulation game. Its internal coherence could not be randomly cobbled together, excluding an infinite monkey theorem or Boltzmann brain like context. So one would have to ignore the human or AI inventors for it to seem totally independent even in terms of origin -- pretend there's a wall or barrier upholding the appearance of a brute situation.
_

 

[Magical Realist]

I once thought long and hard on where arithmetic gets its sense of necessity from. I basically narrowed it down to the phenomenality of the count--of assigning names and symbols to each discrete object in a set or moment of time. We are basically empirically demonstrating thru the counting the consistency of a quantity of objects or moments totaling a specific and unvarying sum--an operation that is irrelevant as to what objects we are counting. Thus 2+2=4 is empirically established as true of necessity, even when stripped of all reference to counted objects/moments. The necessary "lawlike" nature of such equations applies universally to all countable discrete objects. So it does not rely on the objects themselves to be true. It is true unconditionally and universally as a pure abstraction, felt to be true in itself without need of empirical exemplification.

 

[James R]

Regarding 2 + 2 =/= 4, I have two comments

1. Consider modular arithmetic, performed on the number line 1,2,3,1,2,3,1,2,3...

2 + 2 = 1

Where 1 is congruent with 4

This is very similar to the example DaveC gave in his initial discussion with axocanth.

If I recall correctly, axocanth claimed that there are no circumstances in which the statement "2+2=4" would not be true. (I'm sure that axocanth will correct me if my recollection about this is wrong.)

DaveC gave a counter-example where, using base 3 arithmetic, the statement "2+2=11" is true, this refuting axocanth's claim. And, by the way, in that system, the symbol '4' is literally meaningless, which means that the statement "2+2=4" doesn't even make sense, let alone state a truth.

2. More fundamentally, how is mathematical necessity (or any kind of necessity) discerned?

By the usual methods of proof used in mathematics. We start with some simple axioms (which are assumptions), and then apply agreed-upon operations to deduce logical consequences from those axioms.

Mathematicians, by the way, very often make a distinguish between necessary conditions and sufficient conditions for a mathematical statement to be accepted as true.

For instance, a sufficient condition for the statement "2+2=11" to be true is that we're working with base-3 arithmetic.* However, that is not the only way the statement could ever be shown to be true, so the base-3 condition is not a necessary condition for the statement to be true.

Mathematics is a formal system. The starting point is to agree on a certain set of axioms. We can only start to prove statements in mathematics after we have basic agreement about the axioms we're allowed to assume are true.
---

I note that axocanth's response to DaveC's counter-example, in effect, was to try to insist on one particular set of axioms. On the assumption that his preferred set of axioms is the only possible set, axocanth then argued that only the "language" of the statement "2+2=4" changes when we write "2+2=11", and nothing else.

This argument relies on the true but unstated further assumption that DaveC's and axocanth's systems share a certain set of axioms. For instance, we might appeal to real-world experiences, like two sheep added to two sheep results in four sheep. But this is not a formal axiom of any mathematical system. It's also an irrelevant argument, because clearly DaveC's examples shows that even if he and axocanth share assumptions based in real-world experience of sheep and the like, DaveC has still demonstrated that, if we add the additional axiom "we're going to work in base-3 arithmetic", the statement "2+2=4" is not only wrong, but actually meaningless.

As others have pointed out, this is the danger of making blanket claims that start with things like "Under no circumstances could it ever be the case that X is true." Chances are high that you haven't thought through all the circumstances that might render X true and therefore render your claim false. axocanth didn't think through his claim and he was caught out. Since then, thousands more words have been spilled on this topic in this forum, even though the relevant concepts here really aren't so difficult.

---
* We also need to include the more basic set of basic axioms concerning the meanings of the symbols '1', '2', '+', and "=".

 

[DaveC426913]

Thanks James, for saying more eloquently what I have had to say at least five times now, across several threads.

There is a lot of Trumpery going on: "Deny. Deny. Deny" and "Keep telling the lie until the truth is lost." It's odious.