Flavors of Facts
Date: 10/03/2003 at 13:58:14 From: Karen Subject: Is 1 + 1 = 2 an actual fact? Is it a fact that 1 + 1 = 2? I have seen your proof using the Peano postulate. Is the postulate a hypothesis which is unproven, or is it proven, i.e., a fact? For example, 1 + 1 = 10 in base 2. So is the value of 1 + 1 open to interpretation? I think I find some of the terminology confusing, e.g., what do we really mean by the terms 'fact', 'premise', 'assumption', 'axiom', 'postulate', and so on?
Date: 10/03/2003 at 16:46:46 From: Doctor Peterson Subject: Re: Is 1 + 1 = 2 an actual fact? Hi, Karen. You'll have to decide for yourself what you mean by "actual fact"; philosophers can have trouble pinning that down! But consider that every thought has to be based on something else; there has to be some starting point, since you reason ABOUT something. So in order to say something is true, you have to believe that its premises are true first. That might be an observation (but how do you know that your observations are true?); or it can be an "assumption", something we take as true as the basis of our reasoning. That is how we think of math: we choose some set of axioms (or postulates, which are the same thing) and definitions as our starting point, the things we are thinking about. We can choose different axioms and come up with different mathematical systems (such as Euclidean or non-Euclidean geometry). But once we choose them, we consider them to be true -- within the particular system we are working on. An axiom or postulate can't be proven, since there is nothing before it on which to build a proof; it stands at the base of the mathematical system that is built on it. So it is an assumption. But that doesn't make it untrue; it is the truest thing there is _within that system_, the basis of the whole construction. Outside of that system, there is really nothing to tell you whether it is true or false. But we are not "assuming" something about some entity outside of the system, that might really be true or false; rather, we are "assuming" something only in the sense of deciding what it is that our system is about. But your example of 1+1=10 is not really an illustration of any problem with axioms and assumptions. It is nothing more than notation: the numeral 10 in binary is just a different way to WRITE the number 2. The fact you are stating MEANS exactly the same thing as 1+1=2. On the other hand, there is a system (modulo 2 arithmetic) in which 1+1=0. That is no less true than 1+1=2, within its system; but the meanings of 1, +, and = are different than in normal arithmetic. We are talking about different things, based on different definitions. One is a fact about integers, the other is a fact about modulo-2 numbers. One doesn't contradict the other; they just live in different worlds of thought, which are built on different definitions and axioms. So to answer your basic question, yes, 1+1=2 is a fact--given that 1 and 2 refer to the integers 1 and 2, and that + and = have their normal meanings. All the axioms and definitions on which the real number system are based are assumed when I say that! If you don't make some such assumptions, then "1+1=2" has no meaning; it is just a string of symbols on your computer screen, and can't be said to be either true or false. Here is something to read about these ideas: The Role of Postulates http://mathforum.org/library/drmath/view/62560.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 10/03/2003 at 17:45:37 From: Karen Subject: Thank you (is 1 + 1 = 2 an actual fact) Dear Dr. Math, Thanks for that excellent explanation. It has made things much clearer in my mind. In your explanation you make reference to "premises" and "assumptions". Are these the same thing, or are there subtle differences between the two? Regards, Karen
Date: 10/03/2003 at 22:31:12 From: Doctor Peterson Subject: Re: Thank you (is 1 + 1 = 2 an actual fact) Hi, Karen. Well, since I was using the words in an ordinary English sense, and not as math jargon, let's see what a dictionary (m-w.com) says about them, and whether I agree: premise 1 a : a proposition antecedently supposed or proved as a basis of argument or inference; specifically: either of the first two propositions of a syllogism from which the conclusion is drawn b : something assumed or taken for granted: PRESUPPOSITION assumption 5 a : an assuming that something is true b : a fact or statement (as a proposition, axiom, postulate, or notion) taken for granted The (b) parts are pretty much equivalent; but as the (a) parts show, "premise" suggests more emphasis on the logical relationship (that the premise comes before something derived from it), while "assumption" puts more emphasis on the fact that it is not derived from anything else. And I think that's about what I had in mind. They're two halves of the same fact, that something is assumed without proof, so that something else can be proved. It's nice that lexicographers spend time thinking about these distinctions, so we have a head start! If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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