A reader sent me a link to a comment on one of my least favorite major creationist websites, Uncommon Descent (No link, I refuse to link to UD). It’s dumb enough that it really deserves a good mocking.
Barry Arrington, June 10, 2016 at 2:45 pm
daveS:
“That 2 + 3 = 5 is true by definition can be verified in a purely mechanical, absolutely certain way.”This may be counter intuitive to you dave, but your statement is false. There is no way to verify that statement. It is either accepted as self-evidently true, or not. Think about it. What more basic steps of reasoning would you employ to verify the equation? That’s right; there are none. You can say the same thing in different ways such as || + ||| = ||||| or “a set with a cardinality of two added to a set with cardinality of three results in a set with a cardinality of five.” But they all amount to the same statement.
That is another feature of a self-evident truth. It does not depend upon (indeed cannot be) “verified” (as you say) by a process of “precept upon precept” reasoning. As WJM has been trying to tell you, a self-evident truth is, by definition, a truth that is accepted because rejection would be upon pain of patent absurdity.
2+3=5 cannot be verified. It is accepted as self-evidently true because any denial would come at the price of affirming an absurdity.
It’s absolutely possible to verify the statement “2 + 3 = 5”. It’s also absolutely possible to prove that statement. In fact, both of those are more than possible: they’re downright easy, provided you accept the standard definitions of arithmetic. And frankly, only a total idiot who has absolutely no concept of what verification or proof mean would ever claim otherwise.
We’ll start with verification. What does that mean?
Verification is the process of testing a hypothesis to determine if it correctly predicts the outcome. Here’s how you verify that 2+3=5:
- Get two pennies, and put them in a pile.
- Get three pennies, and put them in a pile.
- Put the pile of 2 pennies on top of the pile of 3 pennies.
- Count the resulting pile of pennies.
- If there are 5 pennies, then you have verified that 2+3=5.
Verification isn’t perfect. It’s the result of a single test that confirms what you expect. But verification is repeatable: you can repeat that experiment as many times as you want, and you’ll always get the same result: the resulting pile will always have 5 pennies.
Proof is something different. Proof is a process of using a formal system to demonstrate that within that formal system, a given statement necessarily follows from a set of premises. If the formal system has a valid model, and you accept the premises, then the proof shows that the conclusion must be true.
In formal terms, a proof operates within a formal system called a logic. The logic consists of:
- A collection of rules (called syntax rules or formation rules) that define how to construct a valid statement are in the logical language.
- A collection of rules (called inference rules) that define how to use true statements to determine other true statements.
- A collection of foundational true statements called axioms.
)
Note that “validity”, as mentioned in the syntax rules, is a very different thing from “truth”. Validity means that the statement has the correct structural form. A statement can be valid, and yet be completely meaningless. “The moon is made of green cheese” is a valid sentence, which can easily be rendered in valid logical form, but it’s not true. The classic example of a meaningless statement is “Colorless green ideas sleep furiously”, which is syntactically valid, but utterly meaningless.
Most of the time, when we’re talking about logic and proofs, we’re using a system of logic called first order predicate logic, and a foundational system of axioms called ZFC set theory. Built on those, we define numbers using a collection of definitions called Peano arithmetic.
In Peano arithmetic, we define the natural numbers (that is, the set of non-negative integers) by defining 0 (the cardinality of the empty set), and then defining the other natural numbers using the successor function. In this system, the number zero can be written as 
Using Peano arithmetic, addition is defined recursively:
- For any number
,
.
- For any number numbers x and y:
.
So, using peano arithmetic, here’s how we can prove that 
- In Peano arithemetic form,
means
.
- From rule 2 of addition, we can infer that
. (In numerical syntax, 2+3 is the same as 1+4.)
- Using rule 2 of addition again, we can infer that
(1+4=0+5); and so, by transitivity, that 2+3=0+5.
- Using rule 1 of addition, we can then infer that
; and so, by transitivity, 2+3=5.
You can get around this by pointing out that it’s certainly not a proof from first principles. But I’d argue that if you’re talking about the statement “2+3=5” in the terms of the quoted discussion, that you’re clearly already living in the world of FOPL with some axioms that support peano arithmetic: if you weren’t, then the statement “2+3=5” wouldn’t have any meaning at all. For you to be able to argue that it’s true but unprovable, you must be living in a world in which arithmetic works, and that means that the statement is both verifiable and provable.
If you want to play games and argue about axioms, then I’ll point at the Principia Mathematica. The Principia was an ultimately misguided effort to put mathematics on a perfect, sound foundation. It started with a minimal form of predicate logic and a tiny set of inarguably true axioms, and attempted to derive all of mathematics from nothing but those absolute, unquestionable first principles. It took them a ton of work, but using that foundation, you can derive all of number theory – and that’s what they did. It took them 378 pages of dense logic, but they ultimately build a rock-solid model of the natural numbers, and used that to demonstrate the validity of Peano arithmetic, and then in turn used that to prove, once and for all, that 1+1=2. Using the same proof technique, you can show from first principles, that 2+3=5.
But in a world in which we don’t play semantic games, and we accept the basic principle of Peano arithmetic as a given, it’s a simple proof. It’s a simple proof that can be found in almost any textbook on foundational mathematics or logic. But note how Arrington responds to it: by playing word-games, rephrasing the question in a couple of different ways to show off how much he knows, while completely avoiding the point.
What does it take to conclude that you can’t verify or prove something like 2+3=5? Profound, utter ignorance. Anyone who’s spent any time learning math should know better.
But over at Uncommon Descent? They don’t think they need to actually learn stuff. They don’t need to refer to ungodly things like textbook. They’ve got their God, and that’s all they need to know.
To be clear, I’m not anti-religion. I’m a religious Jew. Uncommon Descent and their rubbish don’t annoy me because I have a grudge against theism. I have a grudge against ignorance. And UD is a huge promoter of arrogant, dishonest ignorance.
it doesn’t seem to me you’re doing math here, but philosophy. I don’t think validity is what’s being questioned, but soundness.
I don’t think that the bozos at UD know the difference between validity and soundness.
But in terms of soundness: if you accept the premises, and you’re working in a system with a valid logical model, then the proof is sound.
So if you think that the UD folks are questioning the soundness of the argument: we know that FOPL with set theory has a valid model. So the only question of soundness in the premises.
So what premises do you think that the UD people are questioning? Are they rejecting ZF logic? (Please correct me if I’m wrong, but I don’t think we need the Axiom of Choice for finite Peano arithmetic!) Are they rejecting Peano arithmetic as a valid model of the natural numbers? What basis is there for questioning the soundness of basic number theory?
You don’t even need ZF here. I think you’d need pairing, emptyset, finite comprehension, and maybe union, but I don’t see how that’s necessary in this case right now. Definitely no replacement or AC or infinity.
Given my run in with antiset theory creationists I could only see them having a problem with power set, AC, infinity and maybe replacement. Literally none of these are necessary for elementary arithmetic afaik.
“If there are 5 pennies, then you have verified that 2+3=5.”
Suppose I’m out of pennies to pile on one another, but I do have this 1 liter cylinder and decide to pump 2 liters of air into it and then 3 liters of air into it. I measure the resulting volume, and find it is still 1 liter. Why does this count as any *less* of a falsification of “2+3=5” than the pennies count as a verification?
What if I take a liter of water and leave it on the Boston sidewalk in January? Will it not expand when it freezes, since 1+0=1?
Nice semantic games there.
In both of your fake cases, you’re starting with one measurement, and then trying to compare it to a different measurement.
You “pump air” into a volume limited cylinder. What you’re doing there isn’t simple addition. You’re not adding one volume to another; you’re adding one collection of gas molecules to the other. If you actually look at what you’re adding, then you could use that to verify addition. If there’s 2 moles of gas molecules in one container, and 3 moles of gas molecules in the other, when you pump the molecules from one into the other, then in the result container, you’ll have 5 moles of gas molecules.
Similarly, when you talk about freezing water, you’re comparing the volume of a quantity of liquid water, with the volume of that same quantity of water in solid form. You’re not doing something that makes sense to model on addition: you’re observing a state change.
In the pennies verification, you’re working with 5 discrete objects, and those 5 objects go through the entire process without being fundamentally changed.
(I’ll just point out here that it’s interesting that these two comments both come from people who’ve never commented here before, and whose profile links to an empty wordpress blog. Gosh, I wonder where they came from!)
Where did I come from? A quasi-rural backwater in 1999-2000 to a real college where people could show me what Usenet was. A recent deconvert, I gravitated to talk.origins (occasionally posting under a different handle) and basically got a free education, and when the posters I admired there moved to blogging and then RSS became a thing I sort of just kept following them. One by one, some of them dropped off my radar (Shallit) or had greatly reduced output (John Wilkins) or sometimes turned toxic (PZ), but by whatever series of events you’ve managed to evade the entropy of my apathy all this time, blog-migrations and all.
Then I noticed you commenting on some particular bit of idiocy from UD (which I can’t believe is somehow still a going concern), and saw that, while the second part about proofs was spot-on, the first part about verification used an example I thought was a little too hastily constructed.
It still seems to me that my examples and yours are relevantly analogous. The reason they give different results is that what is being tested for verification or falsification isn’t the truth of any statement in pure mathematics, but whether some particular phenomenon should be considered a conserved quantity in our models of that system.
It’s only because it’s so easy to smuggle in our background knowledge about the physics of pennies in piles that it looks like the mathematical claim is being tested, when in fact it is the physical claim that e.g. pennies are not perfectly inelastic when piled, don’t behave like drops of water merging into one, don’t multiply like tribbles etc. If any of those things were observed, we’d revise our model, not our math.
You’re pre-screening your examples so they only come out one way, and I’m pre-screening mine so they come out the other way. It’s not fair to say either set counts as a “test” of some proposition if I decided in advance what conclusion I was looking for!
You seem to agree with this in your reply that volume examples are “not doing something that makes sense to model on addition”. That was my point. The normal operation of an abacus is great to model on addition. But if I slide a bead and it happens to be glued to two other beads, then count the result as three slid beads, the correct response is to say the gluey abacus “doesn’t make sense as a model of addition”. The thought of verifying 1=3 this way is never really going to be in the cards.
It seems Peter Gerdes makes more or less the same point below, although in less pithy fashion. I would also underline my agreement with him that this helps the creationists’ misapprehension about how proofs work not one bit.
Before you can infer anything from that evidence you’ll need to prove (or at least convince your audience) that gas volume is always additive. Now you’ve muddied the waters tremondously by mixing chemistry/physics into what was a simple math question.
But hey, all that really matters is getting your audience enough out of their depth so they can’t argue a point, not staying within your own to keep your reasoning sound, so you should be fine.The strikethrough text is exactly what I object to about UD and their cohort.
The whole argument about how you can’t “verify” that 2+3=5 isn’t really a disagreement about verifiability. It’s part of a smoke-cloud. It’s an attempt to derail the conversation, in order to avoid a discussion about something that they don’t want to really discuss.
I don’t think they’re trying to deliberately derail the conversation. That’s the way they really think about the world; that’s what their mental map looks like. It’s a wrong map with pixies and unicorns and an edge of the world – but that’s what they use to navigate.
I wonder how they could be persuaded to improve it?
To be fair there are actually two different things people mean when they say “2+3=5”.
One is the mathematical claim you mention above. However, this isn’t what most non-mathematicians really have in mind when they make claims about arithmetic.
The one that normal non-mathematicians mean is something like “If I count out 2 objects and then count out another 3 I get a pile of objects that I can count out as 5 objects.” This sounds like it’s the same statement but this is actually a contingent claim about the physical world.
To see how these can come apart imagine a world in which whenever I put a pile with two objects in it next to a pile with three objects in it a sixth object appears out of thin air.
What most people mean when they assert facts about addition is that these facts are descriptively true about piles of objects. They are interested in the applications of addition to the world not the mere definitional claims in mathematics.
So yes, there is an assumption we make when we take Peano Arithmetic to accurately model the contingent facts about how accumulations of objects work in the physical world. Of course this does NOTHING to rescue creationist arguments but let’s acknowledge the full complexity of the issue.
I don’t think that most people mean what you seem to think they mean. But whatever.
Sorry, even after all these years, I still struggle with the wordpress UI!
I don’t think most people mean what you said… I think that most people don’t really think about the difference between math and how math applies to reality. I’m pretty sure that 2+3=5 and “If I have two apples and add them to 3 apples, I’ll have five apples” are equivalent statements to most people.
For the UD guys, though, your argument doesn’t work. Look at the quote: he specifically brings up the set theoretic notion of adding cardinalities, and considers that equivalent.
The generous interpretation of their remarks is an argument that goes back to, I think, Wittenstein. If we had a formal system of mathematics in which 2 + 3 = 5 is false, we’d think there was something wrong with the system, not with mathematics. This is fine as far as it goes, but even this generous interpretation fails to support their claims, because for similar reasons we’d reject any formal system of mathematics in which 2 + 3 = 5 was not provable.
Any chance of a new type theory post soon? maybe HoTT for dummies 🙂
@ijp:
I’m working on it. I’m trying to build up a backlog of a couple of posts before I start running it. So soon.
>>For any number x, x + 0 = x.
>>Using rule 1 of addition, we can then infer that 0+5=5; and so, by transitivity, 2+3=5.
Did you accidentally rely on commutativity of addition here? I mean, your steps are easily modified to instead use 5+0=5, but still, rigour, you know… 😉