Yep. No amount of addition will result in the square root of negative one half. — Pfhorrest
My best attempt: — Michael
You can’t get -1 by adding natural numbers to each other. — Pfhorrest
That's your qualification not mine. I can perfectly well consider - invent - a negative number by considering relations between positive numbers. I have $5; I owe $10 - a perfectly natural situation. What is my net worth at the moment? Less than 1. — tim wood
Yes or course, but that relation there is subtraction, not addition. You have X and owe Y, so your net worth is Z = X - Y. So long as X > Y you can start with natural numbers and stay within them, but once X < Y you have to, as you say, invent a new kind of number.
Likewise with division, square roots, etc. They require you to invent new kinds of numbers, because the kinds of numbers you already had aren’t suitable to solve all such problems. — Pfhorrest
That doesn’t seem in disagreement with my point at all, which is that the naturals aren’t closed under subtraction. You have to invent additive inverses of the naturals, creating the integers, or else some subtractions will not have solutions. — Pfhorrest
Sorry, but I have no idea what you're talking about fishfry. The stuff you claim here makes no sense to me at all. When did I say I was just kidding? — Metaphysician Undercover
You know, ZF is only one part of mathematics. If axioms of ZF contradict other mathematical axioms, then there is contradiction within mathematics. In philosophy we're very accustomed to this situation, as philosophy is filled with contradictions, and we're trained to spot them. So we might reject one philosophy based on the principles of another, or reject a part of one philosophy, and so on. There is no reason for an all or nothing attitude. Likewise, one might reject ZF, or parts of it, based on other mathematical principles. — Metaphysician Undercover
Meta's still playing with rocks while the rest of us have pointy sticks.As it turn out, no. Meta has revealed that one cannot subtract from a whole. Subtraction only works if you have more than one individual. And division leads to the heresy of fractions. — Banno
Thus, some things that satisfy the rules of algebra can be interesting to mathematicians even though they don't always represent a real situation. Arrows on a plane can be "added" by putting the head of one arrow on the tail of another, or "multiplied" by successive turns and shrinks. Since these arrows obey the same rules of algebra as regular numbers, mathematicians call them numbers. But to distinguish them from ordinary numbers, they're called "complex numbers. — QED: The Strange Theory of Light and Matter (Richard Feynman)
I did say "all mathematics." My bad. I meant all arithmetic. And mine is really more a question than a claim arising out the the earlier discussion. — tim wood
Ah! So is Meta's contention simply that Presburger arithmetic is the whole of Mathematics? — Banno
In fact the theory of addition is strictly weaker than the theory of addition and multiplication. — fishfry
Multiplication, then, is something more than just an efficient method of addition? What is multiplication doing that addition cannot do? Or is the question(s) I'm asking too simple to answer. For example, am I assuming something I should not? — tim wood
Can you show me a mathematician who has questioned rational numbers like 1919? — Michael
The only thing I'm giving a shot at is for you to see how the math works. — InPitzotl
There's nothing to make clear to me; this is illusory insight. — InPitzotl
Try this... instead of 1/9, let's do 1/7. Now our description has to change, because we get 0.(142867). So yes, each "time" the machine is forced to "loop back" it's because there's a remainder. But what is the remainder to 0.(142867)? Is it 3, 2, 6, 4, 5, or 1? Note that "each time the machine is forced to 'loop back'" it is because there is exactly one of these left as a remainder. Is there exactly one of those left as a remainder to 0.(142867)? Can you even answer these questions... do they have an answer? I'll await your reply before commenting further. — InPitzotl
But if we can't say which remainder this is, we can still talk about the same thing using an alternate view. Suppose we run our long division program and we're told that the result is 0.125. Then what can we say about the ratios it was dividing? I claim we can say it was dividing k/8k for some k. Now likewise suppose we run our long division program and we're told the output is 0.(142857) using the description given by a symmetric recursion and infinite loops. Now what can we say about the ratios it was dividing? I claim we can say it was dividing k/7k. — InPitzotl
You're denying that we can divide at all.. — InPitzotl
But because you worship the idol of the integers, — InPitzotl
The real discussion then is whether we're doing integral division using decimals or rational division, and since decimals are driven by powers of tens (including powers of tenths), it's immediately apparent it's rational division. — InPitzotl
You've got it backwards. They're derived from the axioms of the system you're using. The axioms define various relationships between undefined terms. The application demands use of an appropriate axiomatic system whereby the mappings of the undefined terms have the relationships described by the axioms. — InPitzotl
Because we define it. Incidentally in terms of application we can use this in arbitrarily complex ways. There are some 1080 atoms in the universe, but we can practically get far smaller than 10-80 by applying arithmetic coding to text. Note also that machines can far exceed what we can do, so the limits of what we can do are not bound by some smallest unit of some extant thing... they're bound by the furthest reaches of utility we can possibly get from machines. We can get much further not limiting our theories in silly inconsistent ways. But even without all of this, just for the math is all of the required justification. — InPitzotl
So I start with the fundamental principle of "pure mathematics", which states that a "unit", as a simple, cannot be divided. — Metaphysician Undercover
I already answered this for you. Your request is outside the range of what I asserted, so not relevant. — Metaphysician Undercover
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