• tim wood
    4.7k
    Question then: there are operations in arithmetic that cannot be done - however inefficiently - by addition?
  • Pfhorrest
    2.5k
    Yep. No amount of addition will result in the square root of negative one half.
  • tim wood
    4.7k
    Yep. No amount of addition will result in the square root of negative one half.Pfhorrest

    By addition I mean a function +(x,y) where x and y are numbers, and in the case of non-integers constructed via addition from numbers. What, in your example, lies outside of and inaccessible by this scheme?
  • Michael
    8.9k
    Yep. No amount of addition will result in the square root of negative one half.Pfhorrest



    Solve for .
  • fdrake
    3.9k


    There're two questions there really.

    One is: can you define all operations in terms of addition? This is the question of whether you can generate a bunch of operations (like +, -, times, divide, raising to a power, taking a root) with one operation? In other words; can every member of a select collection of operations be defined in terms of one of the operations within it?

    For the natural numbers, addition generates (addition, multiplication, raising to a power)

    a times b = a + a + ... + a, b times
    a^b = a times a times ... times a, b times

    Another is: is a mathematical object closed under a select collection of operations?

    Closure of an object under an operation is when you can apply the operation to any appropriate collection of its elements and get a result which is still a member of the object. EG, adding two natural numbers {0,1,2,...} will always get you a natural number, so it's closed under them.

    In order to define a bunch of operations on a mathematical object, you usually have to insist upon the closure of that object under the operations; you need to make sure you can apply the operation to everything and get something familiar out.

    @Pfhorrest's example illustrates that since the natural numbers (or integers, or rationals, or reals) are not closed under the operation sqrt(-x) ( sqrt(--2) isn't rational, i=(sqrt(-1)) isn't real) but they are closed under addition, so addition alone can't generate the operation sqrt(-x) (for every x anyway).

    So I'm shaky on the next bit.

    If you move to the complex numbers, which is closed under that operation, the answer is more tricky (@jgill would know much better than me). In the complex numbers, you can define any holomorphic function as an infinite sum. Since the sqrt(x) function isn't holomorphic; there's a discontinuity at 0;, but f(a,b)=a+b where a and b are complex looks like it is... And the composition of holomorphic functions is holomorphic, it seems like you can't end up with a non-holomorphic function by arbitrarily composing a finite collection of holomorphic functions. IE, you can't get sqrt(x) defined everywhere in terms of addition alone, even using complex numbers.
  • Pfhorrest
    2.5k
    I just gave a bunch of examples.

    You can’t get -1 by adding natural numbers to each other. You have to do subtraction, and then that takes you out of the naturals to the integers.

    You can’t get 1/2 by adding (or subtracting) integers to each other. You have to do division, and then that takes you out of the integers to the rationals.

    You can’t get the square root of 2 by adding (or subtracting or dividing) rationals to each other. You have to take a square root, and then that takes you out of the rationals and into the reals.

    Etc.

    My best attempt:Michael

    Clever! :up:
  • fdrake
    3.9k


    The "solve for" operation means you've already put the inverse function of f(y)=k into the mix, its existence is required for that proof to work; that inverse is sqrt(y).
  • Michael
    8.9k
    IE, you can't get sqrt(x) defined everywhere in terms of addition alone, even using complex numbers.fdrake

    From here:

    Definition.
    an nth root of a nonnegative number a is a nonnegative number x such that xn = a

    And exponents can be defined by muliplications and multiplications by additions.
  • tim wood
    4.7k
    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.

    Square root of 2. 1? Too small. 2? Too big. Hmm. Let's try adding 1/2 to 1. 1 1/5? Still too big, but closer. Hmm. 1.4? Too small, but closer still. And so on. Nothing but addition and multiplication. And what's wrong with this?
  • fdrake
    3.9k


    Yes. That is true.
  • Pfhorrest
    2.5k
    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.
  • Michael
    8.9k
    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

    Negative numbers are not the same thing as subtraction. Negative numbers are defined as the additive inverse of the positive numbers, and then subtraction is defined as adding a negative number to another number.
  • Pfhorrest
    2.5k
    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.
  • Michael
    8.9k
    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

    I thought we were discussing the accuracy of tim's claim that "all mathematics is addition"? I don't know about all maths but that seems to be the case for arithmetic. You start by defining the natural numbers and addition. You then define the negative integers as the additive inverses of the natural numbers (the number that when added to natural number n yields zero). You then define subtraction as adding a negative number to another number. You then define multiplication as repeated addition, and division as the inverse of multiplication. You then define exponents as multiplication, and roots as the inverse of exponents.
  • fishfry
    1.5k
    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

    In the very post I was replying to.

    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

    Fine. Find a statement P such that there's a mathematical proof of both P and its negation. That's the only way you can demonstrate that mathematics is inconsistent. I'm still waiting.
  • tim wood
    4.7k
    the accuracy of tim's claim that "all mathematics is addition"?Michael

    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.
  • Banno
    8.5k
    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.
  • InPitzotl
    276
    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
    Meta's still playing with rocks while the rest of us have pointy sticks.
    Reveal
    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)
  • fishfry
    1.5k
    Meta's still playing with rocks while the rest of us have pointy sticks.InPitzotl

    He sure gets into a lot of people's heads.
  • fishfry
    1.5k
    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

    I know a technical context in which that's not true.

    There's a theory weaker than Peano arithmetic called Presburger arithmetic that allows for only addition. It's strictly weaker than PA; and in fact has the remarkable property that it's logically complete in the sense of Gödel. Every statement in Presberger arithmetic can either be proven or disproven.

    When you add in multiplication, you get PA and that is logically incomplete.

    It's commonly believed that in PA you recursively define multiplication based on addition. But it turns out that at a technical level (which I haven't yet grokked) the particular use of recursion is not strictly within the allowable rules in Presberger arithmetic and so is more folklore than truth. In fact the theory of addition is strictly weaker than the theory of addition and multiplication.
  • Banno
    8.5k
    Ah! So is Meta's contention simply that Presburger arithmetic is the whole of Mathematics?
  • fishfry
    1.5k
    Ah! So is Meta's contention simply that Presburger arithmetic is the whole of Mathematics?Banno

    Regarding @Metaphysician Undercover thought process. From Apocalypse Now:

    Capt. Benjamin Willard: They told me that you had gone totally insane, and that your methods were unsound.

    Colonel Kurtz: Are my methods unsound?

    Capt. Benjamin Willard: I don't see any method at all, sir.
  • tim wood
    4.7k
    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?
  • fishfry
    1.5k
    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

    I have not as yet acquired sufficient technical understanding to answer this question. In fact it's a point on which I'm stuck myself. It's a point of logic involving induction. At one point I read the Wiki page and a couple of articles about Presburger arithmetic and thought I had a vague understanding of what was going on; but if I did, it's certainly not stored in my brain cells right now.

    The best place to start is the Wiki page and see if it sheds insight. Also I remember that in my earlier research, I discovered that the Wiki page on the Peano axioms actually hints at the difficulty with defining induction, but again these are memories of a few months ago.

    All I really know is that when you make a recursive definition of multiplication in PA, in the manner I'd always assumed you can do, you are actually adding some kind of secret sauce without realizing it. You can see how little of this I understand.
  • Metaphysician Undercover
    7.2k
    Can you show me a mathematician who has questioned rational numbers like 1919?Michael

    I didn't say any did. I imagine some have, or maybe not. That's not relevant because it doesn't mean that it's wrong for me to.

    The only thing I'm giving a shot at is for you to see how the math works.InPitzotl

    Well, if you think that I haven't already seen how math works, then you're wrong. And as I've already explained, the conclusion I've made from what I've seen is that a healthy dose of skepticism is needed in my approach to mathematics. That is why I've tried to take the discussion beyond rational numbers, to natural numbers, and number theory itself. It appears like you, and most in this forum believe this to be a pointless exercise. That doesn't really concern me. If that's what you think as well, then you're wasting your time here if your true intent is for me to see how math works. I've already seen it. That does not mean that I understand it. If you want me to see why math works, then drop your presuppositions and come to the bottom with me.

    If it's just that I am providing entertainment for you and the others, at least it's of a healthier sort than that provided by the president of the USA.

    There's nothing to make clear to me; this is illusory insight.InPitzotl

    Oh, so you do not see any difference of type between the object we call a pizza, and the object we call a number. That's revealing.

    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

    I can't see how this makes any relevant point. You've just demonstrated another smoke and mirrors method to hide the fact that there is a remainder. If one expression is more vague than the other, then it may or may not be a better way of hiding the fact that there is a remainder. To see what the remainder is at any given time, all we have to do is look to see at what point the machine is at when it loops back. Where's the problem?

    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

    I don't deny any of this, that's how math works, conventions are followed, and that's what convention has us call "dividing". The question is on what principles do we say that the conventions are right or wrong. Do you agree that for any particular way that an action is carried out (an action being the means to an end), in this case a mathematical operation, it is possible that there might be a better way? So even if following the conventions works, there is quite possibly still a better way. We are inclined to say that the conventional way of doing things is "the right way" simply because it is the conventional way, but then what do we say when a better way is shown? One might follow a trail, between the residence and place of work, to and from, day after day, and following that trial always works to get the person where they are going. The person says it's the right way to go to get to and from my work. But that doesn't mean there's not a shortcut. How does a shortcut make the right way into the wrong way?

    You're denying that we can divide at all..InPitzotl

    This is an abysmal straw man.

    But because you worship the idol of the integers,InPitzotl

    You're making the same mistake as fishfry. I do not worship any numbers. In the other thread I was using principles from the rational numbers to attack the real numbers. and for some reason fishfry got the idea that I strongly believed in the rational numbers, just like you think I strongly believe in the integers.

    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

    No, the discussion is whether rational division, as the inverse operation of multiplication, is a true form of division.

    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

    This is the root of the difference between us. You seem to think that mathematics works because people dream up random axioms, then the axioms are applied, and voila, mathematics works. I think that mathematics works because people design the axioms so as to be applicable to the real world. So from my perspective, the real world puts limits on which axioms ought to be accepted. From your perspective, so long as the axioms are coherent and consistent, the mathematics ought to work in the world. Do you see how you are the one who has it backwards?

    So I start with the fundamental principle of "pure mathematics", which states that a "unit", as a simple, cannot be divided. However, I qualify this by saying that whenever the "unit" is applied to the real world, in "applied mathematics", the nature of the object, which the unit represents in that application, determines how the unit might be divided, depending on the object's parts etc.. So the divisibility of the unit is dependent on the object it is applied to.

    You start with the opposite (and what I claim backwards) position, that the fundamental "unit" is divisible any way one can imagine, an infinity of different ways. First, I will argue that this annihilates pure mathematics and number theory, making "one" signify a multitude. Second, I will argue that it leads you to believe, as you've demonstrated in this thread, that any object is divisible in any way imaginable, i.e. an infinity of different ways. So this backward conception of "unit", which you hold, misleads you in this way, actually deceiving you to the point that you will argue persistently that any object can be divided in an infinity of different ways.

    Therefore, the approach which takes as fundamental, that a unit might be divisible in an infinity of different ways, and then might qualify this in application, tailoring divisibility to meet the specifics of the object, is the wrong approach. It is the wrong approach because it has misled you, and others of course, into thinking that mathematicians can produce axioms and the world will exist in the way that the axioms dictate. But when we hold in theory that the "unit" is fundamentally indivisible until its divisibility is proven through practice, we avoid this problem.

    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

    This demonstrates my point.


    Your replies are vague and hard for me to understand. That the procedure proves what the procedure is supposed to prove is not the issue. Of course it will do that or else it would not be an acceptable procedure. The question I thought, was whether there are doubts about the procedure. As I explained above, doubt arises if one believes that there might be a better way. To doubt in this way does not require that the skeptic produce the better way, only that the skeptic demonstrate issues with the accepted way, which might be improved upon.


    Yes, that is what is at issue here, the validity of such inversions, when the inversion turns up something which is outside the rulebook of what it is supposed to be an inversion of.


    I already answered this for you. Your request is outside the range of what I asserted, so not relevant.
  • Banno
    8.5k
    That thought struck me, too.

    I found this, but need a translation.
  • tim wood
    4.7k
    And I wonder what' sso complicated, fishry's notwithstanding. Never mind Presburger. It shouldn't be so complicated.
  • Banno
    8.5k
    Oh, indeed. But the complexity is fascinating.
  • Banno
    8.5k
    So I start with the fundamental principle of "pure mathematics", which states that a "unit", as a simple, cannot be divided.Metaphysician Undercover

    Here's the problem.


    Think I pointed this out before. And I was not alone.
  • fishfry
    1.5k
    I already answered this for you. Your request is outside the range of what I asserted, so not relevant.Metaphysician Undercover

    Don't recall what exactly this was about. Feel our convo is at a plateau at the moment, will be taking a break from our back and forth.
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