# Are emotions involved in conclusions/reaching conclusions in mathematics?

Emotions are not involved in any direct way in our mathematical conclusions (for the conclusions are about numbers, or groups, or vector spaces, or sets, or whatever, not about human things like emotions.) And whether a purported conclusion is indeed a mathematical truth is an objective matter: again, emotions don't come into it. They are not involved in the proofs of the conclusions either (for proofs are deductions from more or less explicit premisses about numbers, or groups, or vector spaces, or sets, or whatever, to conclusions about such things, and still don't mention emotions). And whether a purported proof is indeed a mathematical proof is again objective matter. Does that mean emotions have no place in mathematics? Well, we do think of some proofs as beautiful or elegant or cute. And this, you might well think, is a matter of how we respond -- respond emotionally, in a broad sense -- to the proofs. And what makes a mathematician seek to prove a result in the first place (elegantly or...

# Hello Philosophers Is there any axiomatized theory of arithmetic that is much stronger to be afflicted by Gödel theorems? I've read that there are axiomatized theories that are weaker than the theorem's criteria, i.e not expressive enough, and their consistency is proved within the same theory. I wondered if there would be something like that, which is stronger than the Gödel theorem's criteria for a axiomatized theory.

Gödel's first theorem, with later improvements by Rosser and others, tells us that any theory of arithmetic T that is (i) consistent, (ii) decidably axiomatized (i.e. you can mechanically check that a purported proof obeys the rules of the arithmetic) and (iii) contains Robinson Arithmetic (a very weak fragment of arithmetic) is incomplete. Strengthen T by adding more axioms and the theory will still be incomplete (unless you throw in so much it becomes inconsistent or stops being decidably axiomatized). In sum, you can't "outrun" the reach of incompleteness theorem by going to a stronger theory which is still consistent and properly axiomatized. This is explained in any standard introduction to Gödel's theorems (e.g. in the first chapter of mine).

# I've just read about Cantor'd diagonal argument, and I have some questions about it... Let's say we want to map every real number between 0 and 1 to natural numbers. If I'm not mistaken, that can be done this way: If we have number of form 0.abcdef... (letters stand for decimals, but only some are shown since there is infinite amount of them), then we can produce number N which equals 2^a * 3^b * 5^c * 7^d * 11^e * 13^f * (next prime)^(next decimal). For example, number 0.12 equals to 2^1 * 3^2 (* 5^0 * 7^0 * ...) = 18. Given any natural number N, we can easily determine which real number it represents (if any). My first question is: is all this consistent with Cantor's diagonal argument? (Can both be true at the same time?) Cantor proved there is no one-to-one mapping (not just any mapping), is that important for his result? If yes, it somehow seems intuitive to me, at least at the first sight, that one-to-one mapping can be achieved by simply removing natural numbers that don't represent any real...

But what would an infinite decimal correspond to on the proposed mapping? It may be that every natural corresponds on that mapping to a real between 0 and 1. But you need -- and assert! -- that to every such real there corresponds a natural on this mapping, and that's quite plainly false when you think of the reals with non-terminating decimal expansions (the construction doesn't determine a natural).

# I have been reading discussions on this site about the Principia and about Godel's incompleteness theorem. I would really like to understand what you guys are talking about; it seems endlessly fascinating. I was an English/history major, though, and avoided math whenever I could. Consequently I have never even taken a semester of calculus. The good news (from my perspective) is that I have nothing to do for the rest of my life except for working toward the fulfillment of this one goal I have: to plow through the literature of the Frankfurt School and make sense of it all. Understanding the methods and arguments of logicians would seem to provide a strong context for the worldview that inspired Horkheimer, Fromm, et al. So yeah, where should I start? Do I need to get a book on the fundamentals of arithmetic? Algebra? Geometry? Or do books on elementary logic do a good job explaining the mathematics necessary for understanding the material? As I said, I'm not looking for a quick solution. I...

1. I don't think there is any reason to suppose that learning about mathematical logic from Principia to Gödel will be any help at all in understanding what is going on with the Frankfurt School. (The only tenuous connection I can think of is that the logical positivists were influenced by developments in logic, and the Frankfurt School were concerned inter alia to give a critique of positivism. But since neither the authors of Principia nor Gödel were positivists, it would be better to read some of the positivists themselves if you want to know what the Frankfurt School were reacting against). 2. Of course, I think finding out a bit about mathematical logic is fun for its own sake: but it is mathematics and to really understand I'm afraid there is not much for it other than working through some increasingly tough books called the likes of "An Introduction to Logic" followed by "Intermediate Logic" and then "Mathematical Logic". Still, you can get a distant impression of what's going on...

# I have one question concerning about lines in mathematics. My teacher told me that two lines of different lengths are made up of the same number of points. he told me that if we placed one above the other and join its end points and extend it they will meet at a point (for eg.) R. he told me that we can prove that by joining one point of the longer line to the shorter line and then to the point R and by continuing doing the same. If we do so we will feel that it is made up of the same number of points. But in my view if we place one line above the other and join its end points then both the line would be slanting towards each other (because one is longer than the other). If we remove those points and the line that we joined then equals will be left because we are removing the same number of points. If we continue doing this by drawing parallel lines then both of them will meet at a point on the centre of the shorter line and if we stii continue drawing then the lines will meet at a point such that it...

On the standard account, given two finitely long lines, even of different lengths, their pointscan indeed be matched up one-to-one, e.g. by the kind of projection theteacher indicated. And the possibility of that kind of one-to-one matchingis just what we mean when we say the two lines "contain the same number ofpoints". What makes this possible, despite the different lengths? In part, the fact that there are an infinite number of points along a finite line (the issue we are dealing with here is one of those initially puzzling matters which arises when we deal with the non-finite: intuitions tutored by finite examples can lead us astray). And there being an infinite number of points along a finite line is related to the fact that the points on a line are dense -- that is to say, between any two points, however close together, there is another point . Now, consider a line with end points. Between the left hand end-point a and any other point on the line there is a further point....

# So, it's my understanding that Russell and Whitehead's project of logicism in the Principia Mathematica didn't work out. I understand that two reasons for this are (1) that some of their axioms don't seem to be derivable from pure logic and (2) Gödel's incompleteness theorems. However, particularly since symbolic logic and the philosophy of mathematics are not my area, it's hard for me to see how 1 & 2 work and defeat the project.

I agree with Richard's and Alex's general remarks about "logicism" and what counts as "logical". It would indeed be far too quick to reject every form of logicism just because it makes the existence of an infinite number of objects a matter of "logic". Still, it is perhaps worth reiterating (as Richard indeed does) that Principia gets its infinity of objects by theft rather than honest toil: it just asserts an infinity of objects as a bald axiom rather than trying to conjure them out of some more basic logical(?) principles in a more Fregean way. So I'd still want to say that, whatever the fate of other logicisms, Russell and Whitehead 's version -- given it is based on theft! -- can't really be judged an honest implementation of the original logicist programme as e.g. described in the Principles , even prescinding from incompleteness worries. But for all that, three cheers for Principia in its centenary year!
In the Principles of Mathematics, Russell boldly asserts "All mathematics deals exclusively with concepts definable in terms of a very small number of logical concepts, and ... all its propositions are deducible from a very small number of fundamental logical principles." Principia , a decade later, is an attempt to make good on that programmatic "logicist" claim. Now, one of the axioms of Principia is an Axiom of Infinity which in effect says that there is an infinite number of things. And you might very well wonder whether that is a truth of logic . (If someone thinks the number of things in the universe is finite, are they making a logical mistake?) Another axiom is the Axiom of Reducibility, which I won't try to explain here, but which is even less obviously a logical law -- and indeed Russell himself argued that we should accept it only because it has nice mathematical consequences in the context of the rest of Principia's system. Still, there is some room for...

# In ZFC the primitive "membership" usually has the statement "x is an element of the set y". My question is "is the element 'x'" of a set ever not a set within ZFC?

There's no right answer. Zermelo's original set theory allowed "urelements", i.e. entities in the universe which are members of sets but not themselves sets. Some modern writers use "ZFC" to refer to a descendant of Zermelo's theory allowing urelements. George Tourlakis is an example, in his two volume Lectures in Logic and Set Theory . Some other writers (perhaps the majority) use "ZFC" to refer to the correponding theory of "pure" sets, where there are no urelements and the members of sets are themselves always other sets. Kenneth Kunen is just one example in his modern classic Set Theory . If you are interested in set theory as a tool, then the first line is arguably the more natural one to take. If you are interested in set theory for its own sake, then for most purposes you might as well take the second line (because it seems to make no big difference to the sort of questions that most set theorists are interested in: for example ZFC-with-urelements is equiconsistent with ZFC-for...

# How much do you need to know about mathematics to begin learning about the philosophy of mathematics or, for example, read something like The Principles of Mathematics or Principia Mathematica by Bertrand Russell?

How much do you need to know about science to begin learning about the philosophy of science? Some but not a great deal if you are interested in very general metaphysical questions about e.g. the nature of explanation, laws and causation, and in very general methodological questions about how scientific theories are confirmed and refuted. You'll need to know quite a lot more if you are interested in understanding more specific foundational questions about the interpretation of quantum mechanics or are worrying about the nature of natural selection. Similarly, how much do you need to know about mathematics to begin learning about the philosophy of mathematics? Some but not a great deal if you are interested in very general metaphysical and epistemological questions about e.g. the nature of numbers and the nature of our knowledge of such things (if they are "things"). Quite a lot more if you are bugged by more specific questions about how we are to settle axioms for set theory or to decide...

# How good does one need to be in mathematics to do good work in philosophy of mathematics? Does one need to be able to *do* original math research, or just read and understand math research, or neither? Or does the answer depend on the topic within philosophy of math? If so, which topics are those in which math knowledge is most useful, and in which is it least useful?

You certainly don't need to be able to do original research in maths to be able to work on the philosophy of maths. But you will need to be able to follow whatever maths is particularly relevant to your philosophical interests. How much maths that is, which topics at which levels, will depend on your philosophical projects. For example, compare and contrast the following questions (not exactly a random sample -- they all happen to interest me!): "Is our basic arithmetical knowledge in any sense grounded in intuition?" Evidently, you don't need any special mathematical knowledge to tackle that . "Can a fictionalist about mathematics explain its applicability?" Again, I guess that acquaintance with the sort of high school mathematics that indeed gets applied is probably all you really need to know to discuss this too. "Just what infinitary assumptions are we committed to if we accept applicable mathematics as true?" Here you do need to get more into the maths, and know quite a lot about...

# I have heard that Gödel Proved that Arithmetic cannot be reduced to logic or formal logic. Although I have read explanations which basically state that arithmetic is not complete and thus not definitional like in formal logic, I cannot get my head around how 1+1=2 is NOT reducible to formal logic. This seems like an obvious analytic statement in which "one and one" is the same as saying "two". Can anyone shed light on this?

Well, there is a logical truth in the vicinity of 1 + 1 = 2. Or perhaps better, a whole family of logical truths. Fix on a pair of properties F and G . Then it is a theorem of first-order logic that if exactly one thing is F and one thing is G and nothing is both F and G , then are exactly two things are either- F -or- G . Here the numerical quantifiers 'exactly one thing is' and 'exactly two things are' can be defined in standard ways from the ordinary first-order quantifiers and identity. And the theorem holds whatever pair of properties we choose. This elementary logical result probably captures what is driving your intuition that in some sense 1 + 1 = 2 is "reducible to formal logic". (For a bit more on this sort of thing, see my Intro to Formal Logic §33.3 -- or any other standard logic text!) But all that is quite compatible with Gödel's first incompleteness theorem. For Gödel's theorem isn't about some limitation or incompleteness in our ability to prove ...