I take it, from the way this post has developed, none of the contributors is actually familiar with the Peano Axioms of arithmetic. Firstly, the fact that 1+1=2 can be proved, means that it is a theorem, and therefore MUST be proved - ie, it is not an axiom. Secondly, the proof that 1+1=2 is actually much simpler than they imagine. In its short form, it goes as follows:

1+1 = 1+S(0) = S(1+0) = S(1) = 2

- where the S function = "the successor of".

In its long form, of course, it would require an exposition of the Peano Axioms, which goes beyond the scope of this post.

"Personal preference" is a deeply unscientific frame of mind, Joseph. You use the method which is most appropriate to the context, as determined by scientific principle.

In what way does all of this differ from Hobbes's theory of psychology?

In logical terms, if you seek certainty, then religion is for you. Alternatively, you could limit your mind to those areas of science which deal in certainty, such as Euclidean geometry and Newtonian physics.

If you have the courage, you may choose to embrace uncertainty. In the history of the Universe, the development of Homo Sapiens is a vanishingly insignificant event. Isn't that an invigorating thought?

I chuckle when people assume that morality is impossible outside of christianity. Did Socrates and The Buddha lack morality? Did Confucius?

I think it would be a good idea to use genetic engineering to make ourselves more intelligent. Donald Trump is my main supporting argument. As to making ourselves more attractive, I can't see it. Ideas of attractiveness are too variable.

Fortunately, God willed that Mercy is a virtue. "Go and learn what that text means, I require mercy, not sacrifice..." (Mathew 9:13). You should consider how this might affect the rest of your argument, and restate it.

There are only two ways to answer the question. Either ask God, or ask the patients in a palliative care ward.

"An apparatus is a way of intervening into nature, a way of bringing about a change,"

I'm sorry, why do you think that? If I observe the motions of the planets around Saturn, does that influence those motions?

Granted, in quantum mechanics, we consider that the act of observation entails interference (cp Heysenburg). But in Newtonian or Relativistic mechanics, we don't normally think this holds.

The initial question reminds me of a Tom Sharpe novel.

"How often do you masturbate each day?
a. Twice
b. Three times
c. More often?"

Presumably, whether a required proof is logical/theoretical, as opposed to empirical, will depend upon whether the original question is essentially logical/theoretical in nature, or empirical in nature. It is entirely unnecessary to involve questions of "significance" or "pessimism", which will only confuse the issue.

Empirically, in the first really scientific piece of medical research carried out in modern times, the British Royal Navy discovered in the late 1700's that an infusion of fresh lemon juice into the grog ration of its sailors would prevent the onset of scurvy. The research was entirely empirical, because they knew nothing about vitamins. Their proofs are still considered valid today because they used all the standard modern techniques such as control groups and allowance for placebo effect.

On the other hand, consider the proposition that 1 + 1 = 2. It has no proof (or even relevance) in the empirical world; its validity is entirely logical in nature, and conditional upon your acceptance of the Peano axioms of arithmetic:

1 + 1 = 1 + S(0) = S(1 + 0) = S(1) = 2

But your question has a very respectable philosophical pedigree. For example, the Ontological Argument attempts to prove an empirical fact (the existence of God) by using purely logical arguments. Most philosophers would now accept that that's impossible. Fermi's Paradox, incidentally, fails for the same reason.

Not at all; I am trying to tread a delicate line between "mathematics" and "mathematical philosophy". Most non-mathematicians, and even many mathematicians, conceive of mathematics as the quintessential, monolithic embodiment of perfect rationalism, the ultimate logical system; and of course, it isn't. There are many logical grey areas, even at basic levels. For example, is 0.9... equal to 1? Or is it the largest real number which is less than 1? There are persuasive mathematical arguments on both sides.

"The wise man doubts often, and his views are changeable; the fool is constant in his opinions, and doubts nothing, because he knows everything, except his own ignorance" (Pharaoh Akhenaton).

"Limiting processes" tend to have a somewhat uneasy relationship with the axioms of Set Theory and Peano Arithmetic which underlie damn near everything about number theory. If you are talking about aleph-null infinities then, of course, every aleph-null infinity has a precise numeric value (though this value is impossible to identify).

I'm not entirely sure I follow.

The numbers >=10 from set A can be mapped exactly onto the corresponding numbers in set B. Both sets are infinite, but when set B is exhausted, the numbers 0-9 in set A are left over with nowhere to go. These numbers must therefore be extra to the infinity of set B.

This is dealt with by Russell in one of the later chapters of "Mathematical Philosophy". I don't have the book ready to hand so I'm afraid I can't give you any more precise reference than that; but it's a golden text - any mathematician who has not read it is not a complete mathematician.

Thanks for your reply, BlueBanana. You understand, I guess, that the arithmetic of the countable infinities does not follow all of the same rules as classical arithmetic? For example, that infinity plus any number equals infinity, but infinity minus infinity may equal any number between 0 and infinity, depending on how the infinities were defined in the first place?

Arguments?

Good question. Offhand I would say that, at least in its very broadest logical sense, an "entity" is any object of thought which can be differentiated from its context or surroundings. It could even be "that which does not exist", because as an object of thought, that can be differentiated from "things which exist".

As part of a logical or philosophical discussion, you might want to postulate a less general definition of what you consider an "entity" to be; but of course, to the extent that you would like your definition to place limits upon the kinds of thing which could be considered as a legitimate entity, you would need to have reasons to justify that definition.

Just now saw the moderation standards poll - a good idea, for sure. Look forward to reading that thread.

Hi Goldenverse,

Please do not ask your readers to examine lengthy references, and give "their thoughts". I can buy a book in a charity shop and do that.

By all means supply a reference but, if you have a specific issue to raise, summarise the relevant points and phrase your question in a way which invites agreement or disagreement supported by critical analysis of the reference. Don't be a lazy thinker!

As an offering: Bacon tended to think that if one could only amass sufficient data, the explanatory hypothesis would become obvious, like the picture in a jigsaw puzzle. I t would drop out under its own weight, so to speak. We now know, what Bacon couldn't have guessed, that it isn't as easy as that. There is always more than one explanatory hypothesis. Because "not only is the Universe a queer place, it is queerer than we can imagine".

Please explain the distinction between "Platonist" and "Logician".

PS don't say that you reject an argument because "you don't like it". You must have a rational counter-argument. If some amateur in an internet forum says he rejects Cantor's Diagonal Argument because he "doesn't like it", after a whole century of talented mathematicians trying and failing to prove it wrong, who do you think is going to listen to you?

Trestone,

You present too much argument at the beginning. State your main point in a short paragraph, in terms which are plausible and provocative, and wait for the feedback. Then you can present your detailed arguments.

Thank you for your comments, Srap. Much for me to think about there. Your clarification of the distinction between "subset" and "member of a set" is particularly valuable.

For the moment, I hold to these conclusions:

(1) Neither "even parity" nor "odd parity" fits with the axiom of induction, in the sense that neither can be a property both of n and n+1.

(2) Nor can "parity" (as a general principle) be an essential property of "number", because to say that a number has the property of being either "e" (even) or "not-e" (odd) is trivial and non-informative. (But I need to think more about that; I sense that this argument is probably fallacious).

(3) Consequently the answer to the question whether 0 has even parity, or odd, or no parity, or universal parity, is not self-evident and must be argued.

(4) The usual arguments for the even parity of 0 are facile, self-serving, and question-begging. There are certain mathematical contexts where it is convenient to assume that 0 has even parity, but it does not follow that 0 MUST have even parity. If it does, it must be proved from set theory, or from the axioms of arithmetic, or better still both.

Sorry Srap, I posted that last message accidentally while in draft.

I was working from Russell's articulation of the axiom, which says (Introduction To Mathematical Philosophy, Routledge 2000 paperback edition, p6):

"Any property which belongs to 0, and also to the successor of every number which has the property, belongs to all numbers".

I take this to mean that if P is a property of n, and also of n+1, then P is a property of ALL of the numbers, provided that n=0 is a value of n.

I did miss one important qualification. It is not enough that the property should attach to 0 alone; it must also attach to 1. If that condition is met, it attaches to all numbers, by definition.

The stipulation that the property must belong to 0, as well as to the successor of every [other] number which has the property, I assume is designed to eliminate trivial properties, such as ">10", which clearly could not apply to all of the numbers.

I am not sure I can agree without further clarification, Srap.

The key word is "implies". What does it signify exactly? Does it mean "logically entails", or something less?

If we take it to mean "logically entails", and n=0, then it follows that all of the posterity of 0 has all of the properties as 0, insofar as these are essential to the definition of "number".

There are some properties of numbers which cannot be hereditary in this way, for example, the property of being <1, which applies only to 0 with no implication that it could apply to any of its successors. But such a property cannot be a part of the definition of "number", obviously.

Bear with me, everybody, if I'm always about six replies behind; I'm trying to keep up...

I understand the points made about the cardinality of the null set, because of course the null set does indeed contain one member - itself - and I should have kept that more clearly in mind. But my understanding (probably incorrect!) is that the null set is treatable as having cardinality 1 only in the special context of establishing a purely logical definition of the series of natural numbers.

Outside of that context, once we begin to talk about bundles of objects, relationships, and all of that exciting stuff, the cardinality of the null set assumes a value of 0 to make its properties consistent with those which we expect from the series of natural numbers as a whole. But that's something I'm not at all clear about: when the null set should or should not be counted among the cardinality of a set. For example, {x,y,z} has cardinality 3. The null set is not counted. But its power set has a cardinality of 8 and the null set IS counted.

Moving on, is it meaningful to say that we can divide into 2 equal subsets a set which contains only 1 member? And if it is, in the case of the null set, how to resolve the paradox that division by 2 yields 3 subsets of equal cardinality (if we are to stipulate a remainder of 0)? Is it not a condition of even parity that there should be only 2 subsets of the same cardinality?

Zero may well be an even number. But I sense something specious or at least simplistic about an argument such as "0 is even because 0 divided by 2 is 0, with a remainder of 0". It takes too many short cuts. It smells too much of selective attention. The traditional tests of even parity, applied to 0, turn up outcomes with anomalous features which do not occur in the case of any other natural number, and which require rationalisation. The anomalies are too glaring to be self-evidently consistent with the inductive nature of the series of natural numbers, and surely suggest a need for modification or re-definition of the tests.


In parenthesis, may I say, I like this forum. I have tried other forums before with, frankly, mixed results... I thank everybody for the courteous, intelligent and relevant replies posted.

One other question, if I can test your patience a bit further, The Peano axioms (at least as we have them today) tell us that the series of cardinal numbers is generated from 0 and and every number in the series inherits all of the properties of 0 inductively. But this is clearly not the case with even parity. One might reply that it's the property of "parity" itself which is inductive, but there's something uncomfortable about the idea of an inductive property which can, as it were, change its nature from one number to another. Any comments?

Since parity is not mentioned in the axioms, and is therefore not an essential property of a well-formed number, my own instinct is to argue that it's no more than a useful descriptive feature, an artefact of certain kinds of arithmetical operation.

Thanks for that, chaps, you have certainly cleared a few points for me there and given me something to think about. I didn't know about the spellchecker either!

Thanks for your reply, Michael.

I accept its validity, but how does one express this in set theory? Within the context of this discussion, at least, it implies that 0/2 yields the set {0,0}, in the same way that 4/2 would yield {2,0}. This seems non-intuitive (because the null set can contain only one member) and violates the rule against duplication of elements. This latter rule suggests that {0,0} must collapse to {0}, meaning that the act of division has failed - there are not two identifiable subsets in the null set.