Wednesday, January 27, 2010

Are scientists mostly atheists?

This site, which is a Christian apologetics site, says no.

36 comments:

Doctor Logic said...

"It is likely that those who have rejected religious morality (i.e., those who were cohabiting) wanted to justify their behavior by saying that there was very little truth in any religion."

...and that's why they became scientists? It makes total sense.

So, if any of your family members express an interest in a science career, watch out! They're probably evil! Like me! Bwahahahaha!

Anonymous said...

Nah, you're a good guy Doctor "Logic". You recognize that murder, rape, incest, etc are intrinsically immoral. Right?

Besides, considering the piece argues *against* the claim that most scientists are atheists, and in favor of the claim that the scientists who are atheists are so for reasons that have little if anything to do with science itself, your reasoning goes right down the crapper. As usual.

Ever consider the name "Janitor Logic"? ;)

Doctor Logic said...

Hey Anonymous,

The rate of atheism among scientists is no greater than the average? Was that your take from the article?

BTW, while we're on the topic of morality, who do you like more:

(a) the killer who thinks that murder is intrinsically good,

(b) the killer who thinks morality is subjective, and who subjectively feels that murder is preferable

or

(c) the killer who thinks that murder is intrinsically evil, but decides not to be good, and commits murder.

?

Anonymous said...

Janitor Logic,

I opt for D) The repentant killer who realizes (not thinks, realizes) that killing is wrong, and recognizes the value and sense of avoiding wrongs.

Or lacking that, A), behind bars. At least someone who recognizes the existence of intrinsic goods of any kind is further along a path of reason and logic.

My take from the article was that scientists who happen to be atheists seem to arrive at that conclusion for reasons that have little, if anything, to do with science. And that there's quite a number of theistic or religious scientists.

Do you really think you, someone who expressly rejects that there's such a thing as the truly good or truly evil (not even agnostic: outright rejects) should be prattling on and implying what good person you are? Really, think before you type.

Doctor Logic said...

Well, gee, Anonymous, you seem to have a reading comprehension problem:

It is true that scientists believe less in the existence of God than the general population of the United States.

Got it?

And, if you want to claim moral realism, give us a reason. Hell, give yourself a reason. A reason that doesn't boil down to "I feel X is REALLY REALLY bad". We're waiting...

P.S. You'll be famous if you can do it.

normajean said...

Dr. Logic,

Don't impose your epistemology on me. It's truncated and the cost of rejecting it is low. haha

Mark Frank said...

Bizarre little article. Having discovered a number of different correlations between lack of religious belief other factors - any of which could be causes or not - it announces:

"It is likely that those who have rejected religious morality (i.e., those who were cohabiting) wanted to justify their behavior by saying that there was very little truth in any religion."

Does the writer seriously think that any significant number of educated Westerners in the 21st century think that cohabiting is in any sense immoral or irreligious? This went out about 50 years ago.

Anonymous said...

@Dr. Logic:

"And, if you want to claim moral realism, give us a reason. Hell, give yourself a reason. A reason that doesn't boil down to "I feel X is REALLY REALLY bad"."

The "I feel X" argument can fit nicely into a realist account of moral values. Supporting arguments can be construed along the lines of argument usually used for mathematical platonism:

(1) The city X is bigger than the city Y

(2) The number X is bigger than the number Y

(3) Action X is morally better than than action Y

It seems all three sentences have a truthvalue and the same prima facie structure. It would be a truly elegant theory that would allow us to interpret (1)-(3) as of the same logicogrammatical form, namely

(4) There is an X with a propery a that stands in relation R to a Y with a property b.

Moral realism allows this for (1) and (3), mathematical platonism for (1) and (2).

As Benacerraf imho rightly points out in his essay "Mathematical Truth" such realist accounts usualy have a epistemological problem. Acutally Gödels mathematical epistemology sounds fairly similar to the "I feel x" argument:

"But, despite their remoteness from sense experience, we do have a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true." (What is Cantor's Continuum Problem?", in Benacerraf and Putnahm, p. 271)

Is this a problem for mathematical or moral realism? I think only if we presuppose a causal theory of knowledge. I think such "realisms" work fine with some sort of reliabilism or proper function theory of knowledge. Therefore "I feel X" may have a place, given the proper theoretical background.

Btw, I'm not sure whether I'm a moral realist, I just think there are some interesting arguments for that conclusion.

Mark said...

How'd we get on the subject of moral realism here?

One thing I wasn't clear on was whether agnostics were being represented in the "disbelief in God" category. My suspicion is that they weren't, because I've seen figures thrown around saying that only ~40% of scientists actively believe in God. (And a ridiculously low number of members of the American National Academy of Sciences actively believes in God.)

Mark said...

Also, for good measure, I somehow feel the need to point out the irony of the article's allegations of wishful thinking...

Doctor Logic said...

normajean,

Yeah, having epistemological standards can be a drag sometimes.

Doctor Logic said...

Anonymous,

If (3) is true, there need not be any subjective facts in any realm. I can make an argument for musical or gastronomic realism with equal force.

Also, there's evidence that (1) and (2) are the case whether or not we are paying attention. They make predictions other than "this city feels bigger than that one".

However, the same is not true of (3). Morality predicts how we feel, but little else. Suppose society was very evil. What predictions can we make from this, apart from our feelings will be hurt.

Human morality does have predictably adaptive benefits in human evolution. You could say we predict that human groups (of a certain size) will survive better if they follow certain moral rules. But Lion morality involves killing cubs, and that's what works for lions. We would have to redefine morality in order to construe such facts as evidence for the reality of (3). Morality isn't about survival, unless we redefine it. Survival is survival. Morality is something else.

Jeremy said...

I can't find it at the moment, but I read a recent sociology study that found that most scientists tended to keep whatever belief system they had prior to becoming scientists (with the exception of Jewish scientists apparently). It basically asserted that religious preference seemed to have very little to do with scientific understanding.

That said, I think the common line that people choose something other than God simply because they want to justify some sinful behavior is pretty weak. Assigning intent to people's actions is dangerous at best.

Anonymous said...

"If (3) is true, there need not be any subjective facts in any realm. I can make an argument for musical or gastronomic realism with equal force."

Of course not, I never said that. (I think you mean objective facts). But I needs objective facts if the sentence has the logicogrammatical form of (4). Now do we want to analyze it in the form of (4)? It would be a simple solution and would fit nicely with what most of us think about moral statements.

As for the rest of your arguments: they are obviously question-begging.

"Also, there's evidence that (1) and (2) are the case whether or not we are paying attention" simply states the thesis of moral relativism.

"Human morality does have predictably adaptive benefits in human evolution." This seems to be true, but is irrelevant. Mathematical skills have also such adcantages, yet you agree that they are on a different level than moral skills.

Doctor Logic said...

Anonymous,

"Human morality does have predictably adaptive benefits in human evolution." This seems to be true, but is irrelevant.

Agreed.

Mathematical skills have also such advantages, yet you agree that they are on a different level than moral skills.

But mathematics is always contingent upon axioms. Choose different axioms, and you objectively get different theorems (if any). The sum of the internal angles of a triangle add to 180 if I pick Euclid's axioms, but not if I choose alternative axioms.

The analogy of this in moral reasoning would be something like "If we assume Catholic moral axioms, then it is objectively true that murder is wrong." This would be a true statement, but it is irrelevant to the question of moral realism. What moral realists need is a way to uniquely fix their axioms, and that's what's missing from their program.

In physics, we're trying to find the axioms of nature by working back from the theorems (the effects). Once we have a candidate set of axioms, we test our theory by working forward again and predicting the future.

However, this doesn't work with morality because moral axioms ("oughts") are typically defined to be independent of what "is". If I am kind and generous, and I subsequently suffer misfortune, that proves nothing about whether generosity is good or evil.

Blue Devil Knight said...

Mark asked:
How'd we get on the subject of moral realism here?

Welcome to Dangerous Idea. No matter what the topic is, the comment threads will eventually become about metaethics.

Blue Devil Knight said...

On the excellent question of how they determine religiosity (e.g., are agnostics considered nonbelievers?). At the site Victor linked to they only report the 40.8% of physicists who say there is no God. However, if you look at the original paper, an additional 29.4 chose in the survey: "I do not know if there is a God and there is no way to find out."

So that makes over 70% either atheist or agnostic. 57% of physicists have attended no religious services in the past year.

The site Victor linked to is being misleading, or they just don't understand how to read/interpret a science paper. Neither would surprise me.

The original paper is here.

As we'd expect in the paper, they say, "[R]eligiosity in the home as a child is the most important predictor of present religiosity among this group of scientists."

Not a surprise. Scientists display the same biases as everyone else, though at a much lower rate.

Anonymous said...

Doctor Logic

"But mathematics is always contingent upon axioms. Choose different axioms, and you objectively get different theorems (if any). The sum of the internal angles of a triangle add to 180 if I pick Euclid's axioms, but not if I choose alternative axioms."

This is true, but we don't pick the axioms arbitrarily. We need the Peano-Axiom for almost everything and we think it is true. How do we know? We call them self-evidently true, but what that means we don't know. How is it possible that they are true simpliciter, true without further qualification? What determines their truth-value? The antirealist has no answer to this.

Then there is the problem of Gödels incompleteness theorem: There are propositions about systems that are self-evidently true but which cannot be derived formally. This again is a serious problem for the antirealist.

It seems to me that the antirealist has a lot of work to do before he can claim any plausibility.

How is the situation in ethics?

"The analogy of this in moral reasoning would be something like "If we assume Catholic moral axioms, then it is objectively true that murder is wrong." This would be a true statement, but it is irrelevant to the question of moral realism."

It seems, like in mathematics, that there are true axioms, we don't pick our axioms arbitrarily. The antirealist might answer, that we pick them according to our feelings. But this doesn't help, since the same holds for mathematical axioms and yet most people agree that they can be true. So you certainly need a different argument against moral realism.

"However, this doesn't work with morality because moral axioms ("oughts") are typically defined to be independent of what "is"."
I think most moral realists think that there are ought facts. Whether they supervene on "is" facts probably varies in different realistic theories.

Mark Frank said...

We need the Peano-Axiom for almost everything and we think it is true. How do we know? We call them self-evidently true, but what that means we don't know. How is it possible that they are true simpliciter, true without further qualification? What determines their truth-value? The antirealist has no answer to this.

And does the realist have an answer as to how we know? Get back to Wittgenstein. If we stop thinking of mathematical statements as descriptions of some abstract world of numbers but rather as useful tools then the problem goes away. Numbers are really useful for counting things - but they don't have to exist independently of the things.

Something similar applies to moral statements. Stop thinking of them as descriptions with truth values and rather treat them as moves in our moral life. Call this subjectivity if you like - it doesn't make them unimportant or rule out reasoning or appeals to common ground.

Anonymous said...

"And does the realist have an answer as to how we know? Get back to Wittgenstein. If we stop thinking of mathematical statements as descriptions of some abstract world of numbers but rather as useful tools then the problem goes away. Numbers are really useful for counting things - but they don't have to exist independently of the things."

I agree that the epistemology of the realist is the tricky part. But while this is an interesting challenge for the realist, the antirealists position is in serious trouble because it can't account for the phenomenon of Gödels incompleteness theorem. How can we know the truth of propositions about a system that can't be proved within that system? According to formal theories no such knowledge can exist, but yet it does. To many this is close to a reductio. I'm not sure Wittgenstein knew of this problem. And then there is the problem of choice of axioms.
The antirealist or formalist account of mathematical truth sounds attractive as long as you don't think about those problems too much. I think this is the reason why there are many more platonists in philosophy of mathematics than elsewhere.

Blue Devil Knight said...

Anon, your interpretation of Gödel is off.

Let S be an axiomatic system that is powerful enough to capture basic features of arithmetic (addition, multiplication, and such).

His first theorem is that if S is consistent, then there are true but not provable theorems within S. His second theorem showed we cannot prove the consistency of S (i.e., we can't show the antecedent of his first theorem is true).

And how did he prove his theorems? Using standard mathematical proof methods applied to Gödel sentences. It isn't that we intuitively just know that there are these true but not provable theorems. That is a very common misinterpretation of Gödel that people like to throw around to support Platonism.

For those with a little logic background, this is a great semi-formal intro to Gödel that will prepare you for the Gödelian boners you will surely encounter on the interwebs; it's written by Peter Smith at Cambridge.

Anon also ignored an important question from DLogic. Certain of Euclid's axioms are actually false of the world, yet anon is talking as if axioms in mathematics are intuitively obvious truths. That's a very 18th century way of looking at them, left more than a little egg on Kant's face.

On a related topic, there are many alternative axiomatizations of arithmetic. Do realists prefer the axiom of choice, the well-ordering theorem, or the various other axioms that are logically but not semantically equivalent? Axiomatic set theory doesn't seem to help the realist very much, given the flexibility in our choice of axioms.

Going deeper than ZFC, into logic, it isn't even intuitively obvious that there should only be two truth values, and that isn't even a stated axiom. What of the liar's paradox? Is that true or false? What of paraconsistent logics? There are all sorts of "deviant" logics, and what seems to make many of them deviant is that they weren't developed in the early part of the 20th century. That is, historical accident.

At any rate, not sure how we ended up with this topic. The answer to the question that is the title of this post is 'Yes.'

Anonymous said...

@ Blue Devil Knight:

I'd like to point out that my interpretation is based on Benaccerafs "Mathematical Truth" (f.e. found in Philosophy of Mathematics, An Anthology, [Ed.] Dale Jacquette) and is the recieved view among philosophers of logic and mathematics. I don't want to go into your personal alternative interpretations of his theorem for this is far too off topic and I doubt anyone here has the expertise for this.

" Certain of Euclid's axioms are actually false of the world, yet anon is talking as if axioms in mathematics are intuitively obvious truths."

You misunderstood me. Clearly not all axioms enjoy the same status, there is a branch of speculative mathematics, sometimes called leisure mathematics. It's interesting you mention non-Euclidean geometry, because this is a good example of something that was once considered "leisure mathematics" and is now used in physics. How is this possible? It seems the axioms involve statements about contingent facts. But do all mathematical axioms do? Propably not, it's not clear how the Peano-Axiom could be wrong in some possible worlds.

"Going deeper than ZFC, into logic, it isn't even intuitively obvious that there should only be two truth values, and that isn't even a stated axiom."

Schopenhauers view was that common sense is prior to logic: If our common sense would contradict the laws of some form of logic, we'd abandon this form of logic, not common sense. This seems sensible, since the last tool we have is our mind. So does common sense tell us there are two truth values? Certainly. Almost every philosopher presumes it, even when he's arguing against it.

Shackleman said...

Topic title: "Are scientists mostly atheists?"

Answer: "Yes"

Who cares? The implication in the question suggests that somehow if scientists are mostly atheists then theism is false. I know scientists like to think they are the sole arbiter of truth on *all* things, but they're not (not to mention that they're frequently wrong even on topics that they're supposed to be the experts on).

It irritates me that this question is considered to have any relevance whatsoever. Unfortunately, the masses trust their scientists, more than any other group of "authorities", to tell them what they should and shouldn't believe to be true.

Just as Lewis said, 99% of what we believe is based on authority alone, so one should choose their authorities wisely and with the utmost care. So, before I'm accused of being anti-intellectual or anti-science, I'm not. I trust my scientists on matters of science (not blindly, but trust in them I do). However, in matters of philosophy and theology, I place my trust in philosophers and theologians (also, not blindly).

---------------------------

"Believing things on authority only means believing them because you have told them by someone you think is trustworthy. Ninety-nine per cent of the things you believe are believed on authority. I believe there is such a place as New York. I have not seen it myself. I could not prove by abstract reasoning that there must be such a place. I believe it because reliable people have told me so. The ordinary man believes in the Solar System, atoms, evolution, and the circulation of the blood on authority — because the scientists say so. Every historical statement in the world is believed on authority. None of us has seen the Norman Conquest or the defeat of the Armada. None of us could prove them by pure logic as you prove a thing in mathematics. We believe them simply because people who did see them have left writings that tell us about them: in fact, on authority. A man who jibbed at authority in other things as some people do in religion would have to be content to know nothing all his life."

-C.S. Lewis

Blue Devil Knight said...

Anon said:
[D]oes common sense tell us there are two truth values? Certainly. Almost every philosopher presumes it, even when he's arguing against it.

It isn't obvious at all what to assign to liars paradox type problems. What does common sense say about the truth value of the liars paradox? Is it true, or is it false?

Bivalent logic is one of many logics, perhaps some people are lulled into the sense that it is obvious because it is the milk they drank from their tutor's nipple. There are also trivalent (true, false, or neither) and tetravalent (true, false, both, or neither) logics, and of course fuzzy logic. What logic lets you decide is the best of these options? Or perhaps does it depend on the question, the target domain for which the logic is being used (e.g., to model sentences, claims about the quantum world, claims about medium-sized dry goods).

Perhaps you might say common sense. Again we are left with weird sentences like the liar's paradoxes. It isn't clear common sense is useful here, or that we should expect it to be useful, or that if we have a logical system that bucks common sense we should reject it. Perhaps it will improve on this common sense just as physics improves on folk physics.

At any rate we've hashed over this stuff a lot here so I'll stop as I'm repeating our previous discussions.

Blue Devil Knight said...

Anon said:
I don't want to go into your personal alternative interpretations of his theorem for this is far too off topic and I doubt anyone here has the expertise for this.

I wasn't giving an interpretation of Gödel's theorems, but simply stated them, plus pointed out that his proofs didn't rely on any intuition about the truth of statements in system S. His theorems would never have been published if Gödel had said "look at this it is obviously true but we can't prove it."

He proved, using standard proof techniques, that if you assume S is consistent, then there are true but not provable formulas in S.

This isn't an interpretation of Gödel, but what he proved. You are the one trying to give interpretations of his proofs.

You are also playing the citation game, but that doesn't fly. Making a case for X here in the wild west of the blogosphere involves more than citing what others have said. You have to actually make the case or else it comes off that you don't know what you are talking about.

I know enough about it to know what he proved, and how he proved it. Beyond that is interpretation. If anyone wants to get beyond my blog-comment summary of his two theorems, I mentioned the Peter Smith articles in case you want to check on my accuracy.

I am prone to some excesses on this site, but I am pretty careful about being accurate. Perhaps readers here can let me know if I am being delusional about this. On this question in particular (which I grappled with in philosophy grad school) I am pretty clear on the distinction between what Gödel proved and the panoply of flimsy interpretations of what he proved (despite what anon might say he thinks the "received view" is--perhaps he could improve on my description of what Gödel proved if he is so competent to comment on these matters).

===========

Also, Shackleman is right. While it is an interesting sociological fact that scientists tend to be nonreligious, it isn't clear what this implies for our personal decisions about religious matters.

Anonymous said...

"plus pointed out that his proofs didn't rely on any intuition about the truth of statements in system S"

That's nice of you to point out but nobody made any such claim. I don't quite understand why you mention all this, the theorem itself was never at question. The question was how it relates to the broadly antirealist account of mathematical truth. And there you contradicted the standart view in philosophy of logic and mathematics and called it a "common misinterpretation". If you make bold statements contradicting experts in the field, then you should explain yourself.

Anonymous said...

"(despite what anon might say he thinks the "received view" is--perhaps he could improve on my description of what Gödel proved if he is so competent to comment on these matters)."

You make it sound like this is my own opinion. Since you're obviously not familiar with the literature, here are two famous papers which represent the "received view":

Henry Mehlberg, The Present Situation in Philosophy of Mathematics, Synthese 12, 4 (1960): 380-414

Paul Benacerraf, Mathematical Truh in The Journal of Philosophy, 70, 19 (1973): 661-79

Both papers present this view as the received view.

Anonymous said...

I think I should forward your email to Benacceraf and Mehlberg, I'm sure they'd appreciate your comments on their misunderstanding of Gödel.

Anonymous said...

In case you're actually interested in the issue, here's a more precise formulation which brings out the problem for the formalist well:

"Among Gödel's several 1931 results, one was concerned with the unexpected fact that any nontrivial mathematical theory cannot be prved to be consistent without circularity since its consistency-proof would have to make use of these very properties of natural numbers and of additional, independent assumptions transcending number-theory."

Now here is my old text: There are propositions about systems that are self-evidently true but which cannot be derived formally.

There are propositions about systems (namely about their consistency), that are self-evidently true (one can see the consistency of simple theories without any proof) but which cannot be derived formally (that is within that system).

Blue Devil Knight said...

Anon what are you 12? Go ahead embarrass yourself! Just be sure to include your posts, as that's what I was responding to.

I had just deleted my post to tune it up, but since you posted that silliness I'll just repost and note my changes.
==================

Anon originally said:
How can we know the truth of propositions about a system that can't be proved within that system?

The answer is that Gödel proved it using standard proof techniques.

Then:
According to formal theories no such knowledge can exist, but yet it does.

This is not true, because Gödel proved it using standard proof techniques. There is no magic here, no 'formal theory' that says no such knowledge can exist

And besides, you left out the key antecedent to the conditional in the first theorem. He didn't prove that there are true theorems we can't prove. That's just inaccurate.

Then you go on to say that you have given the "received" interpretation from "experts" in the field, which is just absurd.

Thanks for the fun but I'm done wasting my time with you. Clearly you don't know what you are talking about. [I deleted my previous as I was going to tone down the previous paragraph, but why bother, it is clearly true.]

I'm not saying there aren't interesting issues surrounding Gödel. It is an extremely interesting issue, but it doesn't prove realism is true as you originally stated. He didn't even prove what you originally stated (though it is a common misinterpretation that he proved that there are true but unprovable statements).

[Note added: Again, feel free to send this thread to the experts I'm ready to stand behind everything I wrote. If you think that is a threat that worries me lol. You have done nothing but spout citations (indeed, those are the most intelligent things you said, i.e., suggestions to read people like Benacerraf; everything else you said, the stuff in your own words, the bits that show our actual intellectual mettle? Bologna.

I'm done with you anon. It's turned from potentially interesting diversion from someone who might know what he is talking about, to clarity that it is a time sink with someone who has no clue. Have a good weekend.

I'll respond again this weekend if you write something that belies your patina of ignorance.]

Blue Devil Knight said...

OK Anon you quoted someone and concluded:
"one can see the consistency of simple theories without any proof"

No. That is not true. Indeed, Godel's second theorem is precisely is that it is possible that it is inconsistent (i.e., you can't prove it is consistent).How do you know it is consistent? The whole point is that it is possible that they are not consistent.

Seriously read the Godel papers.

OK I expect doctor logic to step in I'm busy.

Blue Devil Knight said...

Benacerraf is not a realist about numbers anon. Read his work. I had only read godel himself but now reading benacerraf he says he is most sympathetic to fictionalism about numbers.

Look up his stanford interview.

Better yet quit citing people you don't understand and read the original.

My god why do I bother. I thought Ilion was bad.

OK this site is now blocked on my browser. :O

Anonymous said...

You seem really confused Blue Devil Knight. Do you remember how you started this series of misinterpretations?

I posted an argument for moral realism that is similar to an argument for mathematical realism. You pick out a single point of the mathematical argument which shares the resemblance and attack it as if I even tried to defend it. I didn't want to argue for the conclusion, I presupposed it. And I rightly did so because this is the common view.

Anonymous said...

"Benacerraf is not a realist about numbers anon. Read his work."

Of course not, I never said that. He formulating a dilemma! Gödels theorem is involved in one horn of the dilemma.

Mark said...

Anon: "Then there is the problem of Gödels incompleteness theorem: There are propositions about systems that are self-evidently true but which cannot be derived formally. This again is a serious problem for the antirealist."

I think BDK's response is confused here, but I'm nevertheless having trouble seeing this as a big threat to anti-realism about numbers. An anti-realist could say that it's metaphysically possible for the Peano axioms to have a model (a single basketball lying somewhere, with an infinite succession of further basketballs to its left) and conclude (from the Soundness Theorem) that they're consistent. This would require we place some trust in our ability to correctly intuit metaphysical possibilities, as well as our ability to correctly intuit whether a bunch of stuff counts as a model for a given formal theory; but probably the realist would have to endorse the latter anyway, and intuiting merely possible objects seems a lot less suspect than intuiting actual ones.

Blue Devil Knight said...

Mark good point. I'm sure I'm confused.

On your nice basketball example, the question of consistency is tough, because I can imagine that each individual axiom holds (e.g., I call one basketball the '0' ball, define some successor function that jumps to the next basketball, and so on).

Getting a clear hold on the consistency of the system even in this seemingly well-behaved case, is nontrivial (for my brain anyway). And at any rate, we do have proofs of the consistency of other systems (e.g., propositional logic, and I don't understand all the stuff with transfinite induction (Gentzen's proof of the consistency of arithmetic with certain addenda)).

In light of an inability to prove it, as a mathematical question, I stay silent.

(Note above I mentioned some bits of axiomatic set theory as if they were part of the axiomatization of arithmetic I sometimes mix them all together and that's inaccurate. The well-ordering theorem (and axiom of choice) are not part of Peano's axiomatization of arithmetic, but the axiomatization of set theory, and that was a silly mistake).