I may have just used the old 2+2=5 analogy, but I also like this example from the Primate Diaries:
Fundamentalists: believe 2+2 =5 because It Is Written. Somewhere. They have a lot of trouble on their tax returns.
“Moderate” believers: live their lives on the basis that 2+2=4. but go regularly to church to be told that 2+2 once made 5, or will one day make 5, or in a very real and spiritual sense should make 5.
“Moderate” atheists: know that 2+2 =4 but think it impolite to say so too loudly as people who think 2+2=5 might be offended.
“Militant” atheists: “Oh for pity’s sake. HERE. Two pebbles. Two more pebbles. FOUR pebbles. What is WRONG with you people?”
(props to Stephen Wells.)
Well, if there are a lot of believers out there with a lot of weapons and a propensity to use them, I guess I’m a moderate atheist then.
“They have a lot of trouble on their tax returns.”
Hahaha! Good one.
Agnostics: Oh I don’t believe that two plus two equals anything but four. But, I’m not one of those people who swear it’s four.
Mormon: Two plus two equals SEVEN. I had proof, but an angel took it and flew to heaven.
Gould: Two plus two can equal four, but it can also equal five. It really depends on the level of twos you use. We all just have different but non-overlapping values for two.
NOMA: Never Oppose Metaphysical Asininity!
“They have a lot of trouble on their tax returns.”
Is that a Kent Hovind reference? If so, nice, if not, even better :-)
Larouche Youth Movement: To properly appreciate what two plus two equals you must first understand that:
Maybe OT, but CNN is running a Youtube Debate and is asking for people to send in video questions.
http://www.communitycounts.us/debates/
One can vote for or against questions they deem good or bad.
Notice that there are already two good questions on religion/faith. Current question 2 is about appealing to non-religious voters and number 10 is about reason vs. faith.
Both deserve support.
We all know 2+2=5 for very large values of 2
The odds of 2+2 equaling precisely 4 out of all the possible numbers it could equal is less probable than a tornado blowing through a junkyard and assembling a Boeing 747.
Wouldn’t a better analogy be 2.0 + 2.0 = 4?
An engineer loses her wings when you shorten necessary significant digits. Either that or you need to state that you are using the set of all Integers.
Zen Buddhist: We are all one. 2+2 = 1.
Trinitarians: 1+1+1=1
Another good reason to use math analogies.
Math deals with absolute truths (one of the very few fields of science where you can use a binary approach to truth) whereas religion is the belief that something unprovable is absolutely true.
Fundamentalist: 2+2=4 because God says so.
Moderate Theist: 2+2=4 because God set it up so it would.
Moderate Atheist: 2+2=4 because that’s how it works out.
Radical Atheist: Let me introduce you braindead losers to something known as, “number theory”.
Creationist: I fully understand that 1 + 1 = 2 and 2 + 1 = 3… but no matter how much time you give you can’t get to 1000! I fully believe in micromathmatics but macromathmatics is just a religion.
LOL!
As an engineer, I’m in the category of ‘militant’ atheists or ‘moderate’ atheists, but sometimes, I make a sidestep to category ‘moderate’ believer. It depends. Isn’t there a category for people who don’t really care if 2+2 equals four or five?
“It depends. Isn’t there a category for people who don’t really care if 2+2 equals four or five?”
I don’t know, maybe, whatever.
PZ, let me be blunt: if you don’t grasp the difference between mathematics, to which arithmethics belongs, and sciences such as biology and physics, you shouldn’t be teaching one.
The problem here is that rather than doing anything to further your purported goal to have people realize that a wide number of questions are answered far more satisfactorily by science, you’re doing the precise opposite: You are fostering the notion that science is “just another religion” merely refusing to admit it put forth by religious fanatics.
Christian: 2 + 2 = 5, everybody knows Hitler believed that 2 + 2 = 4.
Idist:
2+2=5, but there’s no way it could get that way without a Very Special Helper that we know nothing about, except that he knows math. And that he’s a he. And there’s only one of him. And he set 2+2=5. And he wanted us to know. And no one could have figured it out without him. And he’s very concerned with our sex lives in addition to math. And the guy who figured this out is the Isaac Newton of Mathemagics.
Hankism:
If you believe 2+2=5, Hank will give you a million dollars when you leave town. If you don’t, he’ll kick the shit out of you.
Mushball Moderate:
“Wow, you guys who think it’s 4 are really serious, but so are these other guys who think it’s 5. The only possible answer is 4.5.”
Militant Theist: “I came to know that 2+2=5 by my religion, yet you say you know that 2+2=4 by your “science”, therefore your science is just another religion! Ha!”
I only wish I was exaggerating…
oh man that made me lol
That’s where the cleverness of introducing pebbles into it comes into play. It turns the exercise into an empirical experiment.
I suspect that no one with a scientific background would accuse Myers or Dawkins of fundamentalism and thus fall afoul of Blake’s Law (or corollary or what have you). Only someone whose knowledge of life, the universe and everything is so shallow as to confuse a practical, but unavoidably tentative approach to the world with blind faith could characterize their atheism as dogmatic.
More simply: anyone who invokes “fundamentalist atheism” is ignorant of science, stuck on one side of the divide of C.P. Snow’s “Two Cultures.”
2+2=4, 2×2=4, 2^2=4, 2/2 x 2(2)=4. “Random coincidence” or divine plan?
2 + 1 + ½ + ¼ … looks like it’s going to converge on 4. Maybe you think there’s a chance that an odd denominator will show up eventually. Are you feeling lucky?
Relativists: 2+2=10 if you’re using base 4 or 11 using base 3. 2+2=0 if you’re using vectors and the 2s have opposite vectors. More generally, 0<=2+2<=4 for any vectors for the respective 2s. 2+2=5 if you define the numbering system as 0,1,2,3,5...
#24 @SEF
That’s where the cleverness of introducing pebbles into it comes into play. It turns the exercise into an empirical experiment.
Um, no. It is still based on the simple definition of numbers. The result isn’t “four” because the experiment happened to come out that way, but because the result was defined as being called “four”. More, it’s written 4 only if you are actually using a base 5 or higher system. The reason that it isn’t “five” is plain and simply that four was defined to be the sum of 2+2, or the product of 2×2. If we had called that number “five”, then 2+2 would be 5. It’s plain and simply arbitrary choice. In fact, what’s 5+5? 10? Really? Only if you calculate in the decimal system. Using octal numbers, the result is 12. Why? Because the system is defined in such a way that this is the correct result.
This is the difference between mathematics and other sciences: Mathematics is based on definitions and logical conclusions from those definitions.
#25 Bad jim
More simply: anyone who invokes “fundamentalist atheism” is ignorant of science, stuck on one side of the divide of C.P. Snow’s “Two Cultures.”
I would argue that anyone invoking Snow is stuck on one side. I, for one, have no problem talking with someone in literature research despite being a scientist. The problem is usually with scientists having gross misconceptions about the abilities of science which, almost 13 years after the death of Karl Popper, should have been long buried.
This textbook contains material on 2+2=4. 2+2=4 is a theory, not a fact, regarding the origin of addition. This material should be approached with an open mind, studied carefully, and critically considered.
Let me join you in being blunt: this is humour, you moron.
Thank you MartinM, that I believe is the most polite way I could have put that.
[B]Bog standard, very confused, Is/Ought fallacy lover and moral driveller:[/B]
2+2 must equal 5 because otherwise people would be murdering babies left, right and centre. After all if you cannot admit that, whatever the mundane evidence and facts sa, there is a different level of truth to which absolute morals belong and thus make 2+2=5, then how do you love your wife or manage to stop killing babies. If you think that 2+2=/=5, or even worse 2+2=4, then what stops you from killing babies?
Louis
Pi equals, well something according to a law once proposed.
“Just as people today have a hard time accepting the idea that the speed of light is the speed limit of the universe, Goodwin and Record apparently couldn’t handle the fact that pi was not a rational number. “Since the rule in present use [presumably pi equals 3.14159…] fails to work …, it should be discarded as wholly wanting and misleading in the practical applications,” the bill declared. Instead, mathematically inclined Hoosiers could take their pick among the following formulae:
(1) The ratio of the diameter of a circle to its circumference is 5/4 to 4. In other words, pi equals 16/5 or 3.2
(2) The area of a circle equals the area of a square whose side is 1/4 the circumference of the circle. Working this out algebraically, we see that pi must be equal to 4.
(3) The ratio of the length of a 90 degree arc to the length of a segment connecting the arc’s two endpoints is 8 to 7. This gives us pi equal to the square root of 2 x 16/7, or about 3.23.”
http://www.straightdope.com/classics/a3_341.html
“Lord knows how all this was supposedly to clarify pi or anything else, but as we shall see, they do things a little differently in Indiana. Bill #246 was initially sent to the Committee on Swamp Lands. The committee deliberated gravely on the question, decided it was not the appropriate body to consider such a measure and turned it over to the Committee on Education. The latter committee gave the bill a “pass” recommendation and sent it on to the full House, which approved it unanimously, 67 to 0.
In the state Senate, the bill was referred to the Committee on Temperance. (One begins to suspect it was silly season in the Indiana legislature at the time.) It passed first reading, but that’s as far as it got.
According to The Penguin Dictionary of Curious and Interesting Numbers, the bill “was held up before a second reading due to the intervention of C.A. Waldo, a professor of mathematics [at Purdue] who happened to be passing through.” Waldo, describing the experience later, wrote, “A member [of the legislature] then showed the writer [i.e., Waldo] a copy of the bill just passed and asked him if he would like an introduction to the learned doctor, its author. He declined the courtesy with thanks, remarking that he was acquainted with as many crazy people as he cared to know.””
Then there is this lovely commentary by a biblephile,
“Your response to the question about attempts to legislate pi suggests not only that your scholarship is weak but that you are a heathen. When King Solomon constructed the Temple of Jerusalem, the Second Book of Chronicles, chapter 4, verses 2 and 5, tells us:
“Then he made the Sea [a big tub] of cast bronze, ten cubits from one brim to the other; it was completely round. Its height was five cubits and a line of thirty cubits measured its circumference. It was a handbreadth thick; and its brim was shaped like the brim of a cup….. It contained three thousand baths.”
The ratio of 30 cubits for the circumference to 10 cubits for the diameter “from one brim to the other” of the “completely round” circle gives the value of pi as being exactly 3. Perhaps reliance on the Word of God motivated the Indiana legislators you trashed. You should have checked with the ultimate reference.”
I can’t tell if this guy is being sarcastic or not.
#32 Martin M
Let me join you in being blunt: this is humour, you moron.
Quite right. It’s humourous how people believe that if you compare an apple to an orange, something sensible will come out.
The problem with this “humour” is that it is in line with all the other allegedly non-humorous statements on the blog. And it only works if you accept those statements as true -which I guess you do, but which would also mean that Karl Popper’s life passed you without leaving a trace. In which case I wouldn’t say that you have anything remotely resembling science on your side.
Now, what you have all been waiting for is answersingenesis’ explanation of biblical verses on Pi. Not for the faint-hearted.
http://www.answersingenesis.org/creation/v17/i2/pi.asp
It is all so obvious.
A character in a story claimed to be 1/3 Cherokee, fairly confident that nearly no one would twig to the issue (to which her ready explanation that her father was 2/3 Cherokee).
Funny. I linked to you on my blog. AbstractConcept.blogspot.com
Oliver, before playing the sanctimonious know-it-all role, it’s a good idea to know what you’re fucking talking about.
The reason that it isn’t “five” is plain and simply that four was defined to be the sum of 2+2
Completely wrong. In no form of number theory is 2 + 2 = 4 stated as an axiom. It is derivable from more basic axioms based on the properties of the numbers themselves.
In fact, what’s 5+5? 10? Really? Only if you calculate in the decimal system. Using octal numbers, the result is 12.
The result is the same in both systems. 10(decimal), 12(octal), ten, dix and zehn are all equivalent ways of representing the same number. How something is written is completely irrelevant to what it is.
Likewise God and Allah.
Omigod, the French pray to Dieu, the Spanish to Dios, the Germans to Gott. Though each claims that there is only one deity, they are clearly worshipping mere idols.
Terry Eagleton:
Dawkin’s vulgar caricature of Christian mathematics would make a theology student wince. When it comes to theological-algebra, any shoddy old travesty will pass muster.
What, one wonders, are Dawkins’s views on the epistemological differences between Smitherton’s proof that 2+2=7 and Aldermans theory on the square-root of 2 being 17? Has he read Bentley on Pi being precisely 21, Walton-Smith on the Bessel Function being uniformly zero, or Schmidt on the sine function being its own derivative? Has he even heard of them? Or does he imagine like a bumptious young barrister that you can defeat the opposition while being complacently ignorant of its toughest case?
astronomer: 2 + 2 = 4 +/- 10
(or at least it used to be)
String theorist: 2 + 2 = 0 +/- 1063
Oliver: “Quite right. It’s humourous how people believe that if you compare an apple to an orange, something sensible will come out.
The problem with this “humour” is that it is in line with all the other allegedly non-humorous statements on the blog. And it only works if you accept those statements as true -which I guess you do, but which would also mean that Karl Popper’s life passed you without leaving a trace. In which case I wouldn’t say that you have anything remotely resembling science on your side”
OMG! David? David, is that you or somebody else with the same utter lack of a sense of humor and tendancy to take analogies literally? Could David have a Mini Me?
From my high school days, I remember:
Pi = 3, for sufficiently large values of 3.
-jcr
What is the value of counting pebbles? Can we really “Know” anything? 2, 2, and 4 are arbitrary symbols imposed by narrow thought on a wondrous universe. I can’t even comprehend the limitations implied in “+” and “=”. We have no way of knowing whether there are a constrained amount of pebbles that can be bound up in this “thought experiment.” Everything is an illusion, possibly (but not likely) the 2 + 2 = 4 materialist paradigm is imposed by an “evil demon.” If we only remember that the Universe loves us and wants us to be happy, it doesn’t matter how many pebbles you have, or I have. Or whether we count in base 2. Computers have a right to happiness, too.
Yer all wrong! 2+2=101
2+2=4 may be an alternative to 2+2=5, but is it a POSITIVE ALTERNATIVE? How can we possibly get morality or warm fuzzies from 2+2=4? Wouldn’t everyone be better served if we believed that 2+2=5 for the betterment of society?
Think of the children!
Still a moron, I see. How’s that working out for you?
Gnostic Christian: 2+2=5 because there’s a hidden +1 that is visible only to initiates.
“Oh for pity’s sake. HERE. Two pebbles. Two more pebbles. FOUR pebbles. What is WRONG with you people?”
There’s a t-shirt, right? I need that on a t-shirt.
That is incorrect.
And so is that.
How can we expect people unfamiliar with science to get it right when advocates for science can’t even do it?
Whenever we count, 1 + 1 = 2 and 2 + 1 = 3 et cetera reveals a pattern. This is proof that there is an intelligent designer.
Furthermore, we know that 2 -1 = 1 and 3 – 1 = 1 et cetera, but 1 can’t be reduced without removing its function as a positive number. Irreducible numerology shows that Peanoists are wrong in their materialist religion.
Also, since 1 = 1 always we see that the 2LOT of physics is wrong; entropy can’t possibly increase. At the same time it is obvious that information can’t decrease. This proves that population statisticians are wrong and the numerome that controls cellular algebra is god-… ehrm, design-given and static.
The funny thing is that he is so sure that his view is correct. While others may see mathematics as giving models for those number systems that, well, are observed to work best.
Whenever we count, 1 + 1 = 2 and 2 + 1 = 3 et cetera reveals a pattern. This is proof that there is an intelligent designer.
Furthermore, we know that 2 -1 = 1 and 3 – 1 = 1 et cetera, but 1 can’t be reduced without removing its function as a positive number. Irreducible numerology shows that Peanoists are wrong in their materialist religion.
Also, since 1 = 1 always we see that the 2LOT of physics is wrong; entropy can’t possibly increase. At the same time it is obvious that information can’t decrease. This proves that population statisticians are wrong and the numerome that controls cellular algebra is god-… ehrm, design-given and static.
The funny thing is that he is so sure that his view is correct. While others may see mathematics as giving models for those number systems that, well, are observed to work best.
Uups. The quote was:
Uups. The quote was:
…drops one of the pebbles on the floor and it breaks in two..proudly holds his now 5 pebbles in hand as proof 2+2=5, all thanks to the “Militant” atheist. :)
String Theorist: 2 + 2 = 4, but my pebbles can only be detected at Planck energies, so we can’t yet verify that. (If we’re lucky, however, we might be able to apply arithmetic to quark-gluon plasmas via the AdS/CFT correspondence!)
Greg Chaitin: If you start with the Peano Arithmetic, 2 + 2 = 4 (meaning “successor of successor of zero plus successor of successor of zero equals successor of successor of successor of successor of zero”), but after Gödel and Turing, shouldn’t we really take a quasi-empirical view of mathematics, in which we acknowledge that our choice of axioms is at least somewhat dependent upon what we see in the physical world?
John Conway: Up up plus up up equals up up up up! By the way, have you ever really looked at a brick wall before?
Damn you, Blake Stacey, for making me laugh.
probably a better analogy would be the that the fundies believe that -2 x -2 = -4 because nothing positive can come from negative, while nonbelievers tend to use -2 x -2 = 4. While millenia ago, negative numbers weren’t regarded as legitimate, today few would bother trying to construct an arithmatic where -2 x -2 = -4. It’s much easier and more internaly consistant to have an arithmatic where -2 x -2 = 4.
2+2=5, you bunch of heathens. From the (apocryphal) Book of Inumeratiel, chap. 5:
1 And Inumeratiel went into the valleys to count the flock of his master, Ignorantiath.
2 In the first valley, Inumeratiel did counteth the sheep, and lo, he counted two hundreds of sheep, which he didst count.
3 And lo, verily, in the second valley, he found two hundreds of sheep, which he didst also count in the same manner as the first.
4 And after counting thusly, Inumeratiel returned to the house of his master Ignorantiath and spake to him saying
5 Master, I have counted thy sheep as thou asked. In the first valley thou dost possess two hundreds of sheep, and two hundreds also in the second valley. In sum, thou dost possess five hundreds of sheep.
6 And Ignorantiath was well pleased, and didst gift Inumeratiel with a most comely sheep, that he might know her.
With evidence like this, what more proof could you need?
Hey, PZ, I think you may enjoy this and its associated links.
If 2+2=4, then WHY ARE THERE STILL MONKEYS?
“Dirty” atheist:
I know what you’re thinking. Does two+ two equal four punk, or does it equal five? Well, to tell you the truth, in all this excitement, I’ve kinda lost track myself. But being as this is a .44 Magnum, the most powerful handgun in the world, and would blow your head clean off, you’ve got to ask yourself one question: Do I feel lucky? Well, do ya, punk?
I definitely have to disagree with Oliver on his interpretation of what 2+2 means.
Really each two is the second successor of 0 (and don’t start in on my use of the word second, because I am describing this in English). When you add them together, you have to have as many successors of 0 in the summand as there were in the addends. So there won’t be 5 successors of 0 on either side, there will be 4 on both sides.
Here is how I recall I would have written this in number theory: s(s(0)) + s(s(0)) = s(s(s(s(0)))). Any complaints about my use of numbers in the first paragraph should be resolved by looking at the equation.
Playing games with groups and bases does not change the simple mathematical fact at all.
Not a very useful analogy. A mathematical equation is by its nature falsifiable whereas a religious belief by its nature is not. Believers don’t normally believe in things that are demontrably untrue (well, maybe creationists) but mostly believe in vague concepts for which there is no convincing evidence, just because someone told them about it when they were young and impressionable. Not really the same thing as believing in a demonstrably false equation.
This 2+2 stuff doesn’t work at relativistic speeds, near the speed of 4.
The “secret” is that if you really want 2+2 to be 5, it will happen.
If “2 + 2 = ?” is a question of any real relevance, the answer will turn out to be 42 anyway.
Disprove one of Peano’s axioms, then.
That’s incorrect reasoning, MartinM. The equation must be falsifiable in the sense that it could be shown false by certain results, not that it must be falsifiable in the sense that examination produces those results and makes the statement false.
Our observations are overwhelmingly indicative of the validity of Peano’s axioms. If they were not, we would reject them.
When I was in the first grade, I was paired off with an annoying little twerp who insisted that 2 + 2 = 22. She did all her math facts that way, and when we switched papers to mark them, she “graded” mine according to her little “put the numbers next to each other” method. When I complained to the teacher, the teacher told me that I was going to have to deal with that sort of person my whole life, so I should try to understand. This prepared me, of course, for the sort of tripe I had to deal with from my family and co-workers when I became an atheist.
Both individual equations and individual religious beliefs only occur with a system of assumptions, and these systems can be judged by their consistency. If the assumptions of a mathematical system demand that both 2+2=4 and 2+2=5, then that system is inconsistent. Likewise, there is a long tradition in theological studies of examining whether the assumptions about supernatural entities produce inconsistencies (such as “If God is all good and created everything, how is there evil in the world?”, or “If God is omnipotent, can he limit his actions?”). If one’s system of religious assumptions produce inconsistent beliefs, then that’s a good reason to reject that system.
Damnit, forsen beat me to it.
42 is the only answer you need.
Here’s how you do math, Ma & Pa Kettle style:
According to my Fundamentalist carpool member, all of this is ‘Man’s Math.’ We can’t grasp what He can calculate. If the Bible says 2+2=5, we can only marvel at His wisdom and mathematical skill.
I spell it ‘whiz-dumb.’
Of course, 2+2=5. The animals went into the Ark “two and two“. The Ark was measured in cubits. A cubit is the distance from elbow to fingertip. A hand has five fingers. Could God’s revelatory perfection be more obvious than that?
I will now introduce my idea of a mathematical concept that I believe fits a bit more nicely as an analogy. This concept actually does have near-fanatical opponents with close to the level of young earth creationists’ abilities of reasoning. Plus, I bet a majority of the US population is against this concept as well (like how the majority claims some kind of belief in God).
1.000… = 0.999… (Note: in base-10)
Also known as, “0.999 repeating equals exactly 1 in the decimal number system”.
Hope this thread doesn’t get derailed too much by the (very much for real) “1 != 0.999…” folks.
Fair point. Peano arithmetic may not be falsifable per se, but the proposition that it can be usefully applied to physical entities certainly is.
@hyperdeath #40
Oliver, before playing the sanctimonious know-it-all role, it’s a good idea to know what you’re fucking talking about.
You should heed your own advice first and actually bother to read what others write before you reply to it
Completely wrong. In no form of number theory is 2 + 2 = 4 stated as an axiom. It is derivable from more basic axioms based on the properties of the numbers themselves.
What part of “mathematics is based on definitions and logical conclusions (i.e. derivations) from those definitions did you fail to understand when I wrote it? It is completely irrelevant whether 2+2=4 is stated as an axiom, the numbers are defined in such a fashion that this is the result. That this is define on a more basic level is completely and utterly irrelevant to the fact that it is defined. Which is something completely different than actually concluding it from observation.
@MartinM #50
Still a moron, I see. How’s that working out for you?
Talking to yourself? After all, the one who demonstrated a complete and utter lack of the slightest clue of the workings of science is you. But hey, to each his own. I argue on the basis of the theory of science, you on the basis of insults. Guess who looks like a moron? Once you can articulate yourself like an actually educated person instead of like a high-school dropout, come back.
This reminds me of a mangagement course I heard of that people came away from with the mantra ‘because even 2+2 can equal 5 if you believe it enough’.
This made me laugh out loud. The idea that you could name drop Karl Popper – whose philosophy of science is about as far removed from actual scientific practice as it gets – when accusing someone of not understanding science is endlessly entertaining.
I haven’t actually said anything much about the workings of science.
Maybe true elsewhere, but on this thread, you haven’t much in the way of argument on the basis of the theory of science. I haven’t offered a single argument on the basis of insults. I’ve merely insulted you. Since you’ve presented nothing worth arguing against, but merely ranted on what was otherwise a thread for humour, that seems like a reasonable response.
Still you.
I’m articulating myself like an actually educated person who thinks you’re a moron. Don’t like it, stop acting like a moron.
On the security question answering the Asperger’s poll for 0 + 8 = I answered 08. They didn’t like that, I had to redo my answer, I gave them 8. Shouldn’t this have kicked my poll nunber from 21 to 45?
2+2 = 5 is good if you change your language to say that numbers go 1,2,3,5,4,6,7,8,9 etc. It’s all about framing.
Ken
Zen:
2 pebbles + 2 pebbles = a handful to throw at the novice.
What part of “mathematics is based on definitions and logical conclusions (i.e. derivations) from those definitions did you fail to understand when I wrote it?
The part where it was meant to follow as an apt conclusion to the rest of your post.
It is completely irrelevant whether 2+2=4 is stated as an axiom
Then why did you imply it was? What did you mean by “because the result was defined as being called four”?
the numbers are defined in such a fashion that this is the result.
No they aren’t. You’re trying to redefine addition as some kind of piecemeal operation in which the sum of two numbers is whatever you want it to be. Similarly you’re trying to redefine the natural numbers as completely arbitrary symbols that only exist to satisfy the rules of your pseudo-addition.
In both everyday intuition and the foundations of mathematics, the natural numbers are defined without any recourse to addition. Whether you start by considering a number of apples, or use the Peano axioms, the natural numbers are concepts in their own right.
I… I’m blushing :)
My existence is validated! P-Zed Myers quoted me!
Stuart Pivar:
Draws a series of pictures showing two numeral twos that flop onto their sides, connect end to end, morph into a toroid, and then straighten out into a number five.
poke:
Hey, poke, what are your thoughts about how useful the philosophy of science is to scientists?
And that question can go out to the rest of your as well.
I want to see if my experiences with scientists and science-minded people diverge or converge with y’all’s.
It’s the eagerness of people like Myers to scoff at, and therefore refuse to learn anything from, the tradition that 2+2=5 that bothers me. Among other things, it cuts them off from much of the world’s great literature, art, and music.
Discordianism:
2+2=5. Or maybe 23. Or both.
Rastafarianism:
2+2=…wait, what?
Pastafarianism:
2+2=Fahrrrrrrrrrrr, matey
@Blake Stacey, OM: (#57)
oh, a conway joke! i actually came here to relate a conway story. i once went to one of his lectures in which he proved, mathematically, that 2+2=3 (for exceptionally low values of 2). i forget precisely how he did this, as it was a number of years ago, but it involved changing his premise into a conclusion, or some such strange thing.
what i recall more about that particular conference was breaking one of conway’s famous puzzles.
The most useful book on philosophy of science I ever read is Francis Bacon’s “On the advancement of knowledge.” Should be required reading.
Most other works in the field have struck me as being about as useful as a manual on “How the world should have been explored.”
abeja @62
ITYM “If you doubt 2+2 = 5 is possible, how is it there are PYGMIES + DWARFS??“
Allow me to add:
YA religious nutjob: Of course 2+2 = 5, but the Bible warned us that this would seem like foolishness to those who refuse to receive the Holy Spirit.
Theologian: Arithmetic does not deal with material objects: can you show me a pound of “two” or a liter of addition? Therefore such truths are not subject to empirical verification. And while no serious thinker believes that 2+2 literally equals 5, Aquinas and others have convincingly demonstrated that since our definitions of “2” and “+” are contingent entities, there must be a necessary value lying at the root of the process. After all, if a+b = c, then a+(b-1) = (c-1). The only way to halt this infinite regress is to posit a First Integer, which we shall call 5.
If 2 + 2 = 4, then why don’t we see any giant mammaried mosquitoes?
Caledonian: I think scientists who want to understand science are better off taking a history of science class than a philosophy class. That said, one of the reasons to read philosophy is that we’ve all picked up some unexamined philosophical platitudes along the way (for scientists it’s usually scraps of Hume and Popper), and the best way to subject them to criticism is to read the source (angrily and incredulously). (If you’ve ever found yourself saying “Hume says…” or “if you can falisfy…” then do yourself a favour and read Hume and Popper. Or just stop doing it.)
I think David Stove’s demolishing of Popper in his book Scientific Irrationalism is worth a look for exactly that reason (although ignore most everything else he’s written; especially on evolution). His criticism that Popper (and Kuhn et al) confuse logical and historical theses is on point IMO and he’s amusingly caustic in his approach. I also like the second part of Ian Hacking’s Representing and Intervening where he tries to establish an approach to philosophy of science that focuses on experiment rather than theory (the first part is a somewhat flaky review of other philosophy of science).
This gives me a creepy hive mind feeling. Oh well, I guess that’s what I get for copying cleverness versus coming up with it.
Pardon my mathematical ignorance, but 1.000..=.9 (repeating) would seem to be an obviously false statement. It’s almost 1, but not quite, by an absurdly tiny amount. I agree that for all intents other than unnecessarily complex math it makes no difference, but there IS one, right?
(this is an honest question btw. I like to think I’m a fairly smart guy, but complex math isn’t something I know much about)
@Mechalith’s comment 98: before you get stomped by angry mathematicians, no, 0.99999(recurring) is NOT less than 1, even by an absurdly tiny amount. The infinite recurrence reduces the difference between 0.9999999… and 1.0 to zero. (If you terminated the recurrence at any finite number of 9s, then there would be a difference, but “recurring” mean you never terminate that string of 9s)
A good way to see this is to consider the following:
x=0.99999999….(recurring)
10*x = 9.99999999….(recurring)
(10*x) – (x) = 9x =9.0
x = 1.0
BTW, it’s a bit scary that I seem to have started an internet meme. WIth luck it’ll only be a 8.999999…. days’ wonder.
For what it’s worth, the 1.0 = 0.999… thing became a lot clearer to me when I considered that it’s a logical necessity given that
1/3 = 0.3333…
and
0.333… + 0.333…. + 0.333… = 0.999…
More than anything else, it’s a clear demonstration of how human intuition utterly fails to agree with logic when dealing with infinities.
Thanks for the reply, poke.
***
That’s not quite accurate. The material objects in question are logic states – in this case, in your neural net – and can only be known through empirical verification.
There are, of course, certain additional requirements that need to be met, because “2+2=4” is only true for things that have specific sorts of properties. Two cups of sand added to two cups of water is not four cups of sand-and-water.
Mechalith: I direct you to
http://qntm.org/pointnine
That should settle things. It goes over quite clearly several valid arguments for .9…=1, as well as showing why several objections to it are incorrect.
Stephen Wells:
Memes are, indeed, frightening things to invent! ;-) Yours reminds me of the one about explaining systems of government using cows.
According to horrobin, Stuart Pivar
Fantastic! Let’s see what else is brought to mind by recent events. . .
Wikipedian: 2 + 2 = 4, but the neutrality of this equation is disputed.
Alternatively, 2 + 2 = 4 [citation needed].
(I remember there was a big stink over where footnotes should go in scientific articles, and how many should be required.)
Its ironic how someone brought up PI after some other twit complained we where comparing apples and oranges. See, we don’t know what or **if** PI has a final digit, and there are lots of things in math that build on well established math concepts, which we don’t know yet. Math is a science. He wants math to be something carved by lightning into stone tablets, so he can complain that, *unlike math*, biology and other sciences aren’t. Wrong!!! Most people just can’t fracking understand the parts that are well known, let alone why there are so many unknowns in it. But since everyone can do 2+2 and “presumably” come up with 4, while its not so much true that most people can do even something as simple as keeping a house plant alive, never mind splicing genes.
Its the gap in skills needed, and the overblown assumption of more skill in math than they themselves actually possess, that makes it “look” like math is different than biology. If on the other hand we taught that math was complex and hard, and even the most basic applications of it where above everyone’s head, but biology was so accessible, or simply defined, that two people humping could claim to know how to do it, then the entire argument would be 180 degrees reversed, never mind that they still wouldn’t be any better and biology or math than they are right now.
“What is WRONG with you people?”
That would be “What the FUCK is WRONG with you people?” Remember, we’re militant. We say “fuck” a lot.
Pragmatism: 2+2=4… However, for the right price, we’re willing to negotiate any number you want.
This reminds me of a mangagement course I heard of that people came away from with the mantra ‘because even 2+2 can equal 5 if you believe it enough’.
I think my boss took that course. The textbook is “1984”, right?
Kagehi @ #104,
A clarification: do you claim that we do not know whether or not Pi is irrational or are you claiming that most of us do not have enough mathematical sophistication to know the answer, let alone perform a proof?
We know that pi is an irrational number, and that any digit-based representation of its value must necessarily have an infinite number of digits.
Whether that infinite sequence has a final value is something that, to my knowledge, we do not know how to determine.
Does the International Journal for Approximate Reasoning cover the fundamentalists and moderate believers?
Caledonian,
Unless I am terribly mistaken, an infinite irrational sequence with a final value is a contradiction of terms. By “final value” do you mean a definite real numbered quantity, or final digit in its expansion (contradiction), or something else?
Sorry to be such a nitpick.
I would say that we do not know for certain. There have been a few other cases where numbers where *thought* to be irrational, but they later found that they did have a final result, at like some huge decimal place. Its not *likely* that PI is one of them, but people don’t just keep computing more and more digits just to get more and more digits, they also, in some cases, are trying to determine “if” its really irrational, or if we just don’t have enough digits yet. We just don’t know, unlike in cases like 1/3. So, no, we “assume” that it has infinite digits. Just as we assumed some others did, until someone found out they didn’t.
Its one of those cases where proving its irrational may not be possible, while proving it isn’t is computationally impractical…
Linear infinities can have a defined beginning that continues forever, a defined end that is preceded forever, or known beginnings and endings with an infinite “in-between”.
In base-ten, the first digit of pi is 3. This is known.
It is conceivable that there is a last digit of pi with a specific, definite value – that is separated from the first digit by infinitely many digits. Infinities can have known boundaries and unknowable middles.
To my knowledge, we do not know if there is a means for determining whether pi can be said to have a last digit, and if so, what that digit is.
We do know that pi is irrational, and cannot be completely represented as a series of digits without an infinite number of them.
Kagehi,
What are these cases you allude to? I am genuinely interested as a mathematical hobbyist.
What do you not accept about the proofs of the irrationality of Pi? They follow as irrefutably as any proof by contradiction I have seen. Do you see it’s proof as being an empirical procedure whereby Pi is enumerated and checked for a final digit?
Xanthir (#102),
I like that link you provided, since it gets to some of the main points in a neat and short manner.
Anyway, the reason I brought up the whole 1 = 0.999… thing was because I noticed how creationists/IDists and “1 != 0.999…” folks show the same unwillingness to face basic facts (or, even if accepting the facts, don’t accept the direct logical conclusions from said facts), redefine terms and concepts to suit their whims, show poor understanding of basic concepts in science/math, claim that mainstream science/math is flawed (a.k.a. “I’m not a scientist/mathematician and have not done much/any studying of the field, but I know that [insert widely accepted scientific/mathematical concept] is just plain wrong! My shallow, layman, common-sense knowledge trumps anything these “experts” claim from their decades of studies!”), etc.
I have seen all of the above things in the comments for this thread:
http://tinyurl.com/fn44j
at Polymathematic’s blog, as well as for the follow-up threads. Seriously, the logical fallacies used by these two groups match up pretty well with each other. And unlike the 2+2=5 stuff, this is for real. Here’s a sample comment of what I’m talking about:
“I am not going to pick a side at this point, but will point out holes on both sides, if we go with your logic, and the statement that there is no number between .99999… and 1, we can also assume that there is a number (n) where no number can be found between .99999… and n, we must also assume that there exists a number x, where no number between n and x can be found, taking this logic further, we can assume that since, 1=0.99999…=n=x, all numbers are equal to each other. On the other hand to deny your proof would be to deny fundamental mathematical stuff, which i don’t feel like doing today. Thank you for the thought provoking series of posts all.”
Ow… the stupid… it hurts…
Note: I don’t think people are stupid for not accepting right away that “1 = 0.999…” (I sure was skeptical the first time I saw the equation). What I think is stupid is to be shown the proofs and explanations for the equation, yet STILL insist on denying it based on flimsy reasoning or outright fingers-in-ears singing.
Well, first off, it was over two years ago that I read an article or two on the subject of those numbers, so I don’t even remember what they where exactly. Second, unlike you Johnny Logic I am not a huge math enthusiast, so I might not understand “if” any proofs if its irrationality make sense or not to even try to refute them. lol My central point was that math isn’t exactly all known or all tested, or anything else implied, and there is some allowance for interpretation when it comes to “if” certain things are or are not one thing instead of others in it. Whether or not PI is one I am not in a position to dispute or argue, one way or the other.
2+2=5 once you add sales tax.
Caledonian,
I’m not familiar with the term “linear infinity”, but I assume you mean infinite sequence, or perhaps you refer to intervals on the reals?
Sure, any closed interval on the reals has a known beginning and ending with an infinite middle, but a point, whose expansion is an infinite, nonrepreating sequence is not one of these. I suppose in a sense you can say that 1/3 has a definite end of 3, because it is threes all the way down, if you will. Still, no three is the last three. Pi by contrast is nonrepeating, so there is no identifiable “ending”.
Imagine the decimal expansion of Pi as a series of intervals narrowing downward to the point Pi, represented by a sequence of natural numbers Pi = <3, 1, 4, 1, 5, …> (3 in the first place designating the interval 3.00.. to 3.999…; 1 in the second place ordered with three designating the interval 3.100.. to 3.1999..; etc.). The interval shrinks ever smaller as the expansion is enumerated, converging on Pi. Pi is just the closure of all of these intervals–the limit point of this sequence. What does it mean to say that it has a definite value at the end of an infinite sequence of natural numbers, when its entire definition is in the aforementioned nonrepeating sequence of natural numbers? Pi = <3, 1, 4, 1, 5, …, x>? This would seem to have a different ordinality (ω+1)of all other points on the reals (ω)…
On the bright side, posing this conundrum is a proven method for foiling alien enemies.
Kagehi,
Points taken.
I once heard this comment from an economist in Central Asia: “2+2 = 3 wholesale or 5 retail.”
Oops.. comments ate my ordered numbers. It should have read:
From Precalculus for Christian Schools, a textbook published by Bob Jones University. The book is designed to help students “conform their thinking to biblical precepts.”
http://www.harpers.org/archive/2004/12/0080307
“The concept of limit can be used to illustrate an important truth. Suppose you lived eighty years and there was no life after death; your life on the earth would be 80⁄80 = 1 = 100% of your existence. Now, let’s assume that your life after death was eighty years long: your earthly life would be 80⁄160 = 1⁄2 = 50% of your entire existence. If life after death were 720 years, your life here would be only 80⁄(80+720) = 0.1 = 10%. Now extend it to eternity: [the limit as x approaches]∞80⁄(80+x) = 0. In other words, this life is very insignificant in light of eternity. It is no wonder James said that life is “a vapour, that appeareth for a little time, and then vanisheth away.””
That’s not its entire definition, for one thing. None of the points you bring up contradict the possibility that pi has a definable end.
OK apparently it hates html encodings of ‘greater than’ and ‘less than’ symbols, as well. Sorry for the boring repeats, but I want to be clear:
You are probably tired of the subject, but could you explain how having a different ordinality than all other real numbers if it does (have a transfinite “end”), when it itself is a member of the reals doesn’t show the question of it having a definable end to be problematic?
And, again, what sense “end”? Also, what did you mean by “linear infinity”?
I am just a curious dabbler– these are serious, not sarcastic, questions.
Er, it’s about as certain as anything in mathematics that the sequence of decimal numbers representing pi has no end.
Pi has been proven to be irrational since, I don’t know, like 400BC or something (my copy of “History of Pi” is at home…) Irrational, of course, means that it cannot be expressed as a ratio of any two integers.
Obviously, any sequence of decimal places that terminates is rational. If it terminates at the nth place, it is expressible as an integer divided by 10^n. This is so self-evident I’m not entirely sure whether I’m misunderstanding Caledonian and Kagehi, or if they’re just grossly underinformed.
There’s nothing especially magical about the irrationality of pi — Cantor showed that the irrational numbers outnumber the rational ones by a factor of roughly infinity. (http://www.mathpages.com/home/kmath371.htm) The property of transcendality is a bit more interesting, I don’t think that was proved until relatively modern times. At any rate, for more thoughtful info on this I do not hesitate to recommend Petr Beckmann’s “A History of Pi”
Ah, never mind, I missed comment #113. Still not sure that “the final digit of pi” is in any way a sensible statement, but the concept of transfinites is about the point in higher math where my head starts to melt.
Neither am I. However, I can’t show that it isn’t, and to my knowledge, no one else can either.
Maybe math already has a demonstration that the concept is valid/invalid – that’d be nice. But if it exists, it’s way beyond my limited knowledge of the subject.
Maybe I’m just being ignorant (again), but I’d think the very idea of an infinite sequence having a last anything was self-contradictory.
(thank you, btw, to all of you who provided explainations and proofs of the 1=0.999(recurring) thing. It still makes part of my mind rebell noisily when I think about it, but at least I can see that it IS true and why now.)
“It still makes part of my mind rebel noisily when I think about it”
Me too, and isn’t that interesting? I wonder if the rebelious part is some sort of innate math module (ala Hauser or Pinker), or an early-learned intuitive sense of arithmetic. Or what.
Congratulations to Mechalith on being able to accept logical proof even when it makes your intuition rebel. It’s a rare trait, but vital.
Caledonian, it’s fairly clear by now that you don’t know what you’re talking about re. the “last digit of pi.” The decimal representation of pi continues for ever; there is no last digit. You’re being contrarian for no reason.
Sure it is. The infinite sequence is sufficient to characterize exactly one real number, and that’s pi. It’s a complete definition.
Naive Diarist: 2 + 2 = 4
Daily Kossite: Lutefisk:
1 kg dried fish
100 g caustic soda
30 liters of water
Saw the fish in suitably sized pieces or leave it whole. Put in water. Leave in water in a cool place for 5-6 days if cut in pieces, 8 days if the fish is whole. Change the water every day.
[etc.]
s9:
Great. Now I’ve got that song stuck in my head.
That’s not the only definition of pi – it’s the ratio of the circumference of a circle to its diameter, and the limit of a particular summation of fractions, among other things. The definition is enough to specify pi, but it’s not a complete one.
Bernarda:
Oh, dear Ghu. I thought Harpers was making that textbook up, but it turns out it’s real.
In that sense, a complete definition of pi is impossible; at least, I know of no way to summarize every possible expression to which it’s equal.
But it’s common in math to say “completely defined” when you mean that the definition corresponds to exactly one entity–existence plus uniqueness. In that sense, pi is completely defined by its digit sequence. There’s nothing left for a “last digit” to do.
Blake Stacey, OM: IMO, Gödel’s theorem should show us that fictionalism about math is true, but …
2 + 2 =
some number.
Why do you want to know,
which number?
Are you authorized to know,
which number?
Bank foreign currency exchange:
2+2 = 3 if you are selling,
2+2 = 5 if you are buying.
Cal: Which part of [infinite] did you have difficulty with.
If the series term of Pi in infinite, there is no *last* – by definition. If you’re visual in nature, it’s fractal: zoom in closer & closer & closer and there is more and more and more detail. If you look at a fractal boundary: at a given specific resolution you can say with certainty what the color of a specific pixel (function result) will be…. but all you are doing is truncating at the n’th digit. That still tells you nothing about the complexity of the series beyond that point.