Leibniz on 1+1=2

From Kuo & Joe, here’s an interesting article on different philosophers’ views on what it means for 1+1 to equal 2 and how their concept of divinity plays into their ideas. Leibniz’ view seems the most compelling:

When Gottfried Wilhelm Leibniz, an inventor of the calculus, was asked by one of his students, “Why is one and one always two, and how do we know this?” Leibnitz replied, “One and one equals two is an eternal, immutable truth that would be so whether or not there were things to count or people to count them.” Numbers, numerical relationships, and mathematical laws (such as the law of addition) exist in this abstract realm and are independent of any physical existence. In Leibnitz’s view, numbers are real things that exist in a dimension outside of the physical realm and would exist even if no human existed to recognize them.

I don’t know if the exchange between Leibniz and his student is real or merely apocryphal, but I’d say his view is closest to my own. The article also has summaries on the positions of Bertrand Russell, John Stuart Mill, and John Dewey, all of whom operate from various points on the atheist or rationalist spectrum — and it shows.


Filed under Christianity, Math

19 responses to “Leibniz on 1+1=2

  1. virusdoc

    In a physical sense, this is true. Two electrons have twice the mass and electrical potential of one. A planetary object twice the size exerts twice the gravitational attraction. Two “ATG” codons encode for two methionines, which changes the sequence and function of the resulting function.

    But this doesn’t mean that mathematical concepts transcend physical ones. There is no math without the universe. Math is a beautiful means of describing that which is, not the converse.

  2. virusdoc

    Sorry–“sequence and function of the resulting protein”.

  3. Nadya Fermega

    Similar question why 1 – 1 = 0 but another physical sence that is how to prove it, ho ho ho …..

  4. Likewise, virusdoc, I’d argue that just because math exists apart from the universe doesn’t mean that it can’t describe very well the things that are in the universe. “Supernatural” in that sense does not mean “unnatural”.

  5. Nadya Fermega

    Sorry sir, my simple question is not a joke. it will reveal the secret of science that maybe next time to becomes our living be better. But if we ask to the mathematicians about 1-1 =0, I am sure they will answer with the relaxed that 1 + (-1) = 0. Thinking about my purpose sir. Thx.

  6. Nadya Fermega

    I am interested to post some comments here because I want to inform you about what I’ve read on
    that maybe useful to answer Denaya’s question about how to prove that tan(pi/2)=0/0.

  7. Nadya Fermega

    I am interested to post some comments here because I want to inform you about what I’ve read on
    that maybe useful to answer Denaya’s question about how to prove that tan(pi/2)=0/0.

  8. Mr. Robert, I think miss Nadya is right. Besides we need a specified model for treating some phenomenons such as the global crisis of financial and other modern applications, but we must still solve the mathematics model. I agree that the usual mathematical models are in differential equation (DE) explained by you previously. Although there are several establish methods that has been commonly used to solve the DE, but maybe we still develop alternative way that perhaps provides another solution form but giving more complete explanation about a phenomenon. I take one example that reasonable to be discussed further.

    “The discussion about global warming has been commonly associated to the Planck’s formula of Black Body Radiaton. We know that the Planck Formula for the internal energy of black body radiation is of form :

    U(T) = ћω / [exp(ћω/(kT) – 1]

    But when Einstein verified the phenomenon using harmonic oscillators, the above Planck’s formula can only explain us that the energy difference between the sequence of two energy levels of harmonic oscillator is ћω. But if the internal energy is taken in the following form :

    U(T) = 0.5ћω[ tanh{0.5ћω/(kT) – iП/2} – 1]

    We will also verify that the minimum energy of the harmonic oscillator is 0.5ћω, and justifying that there are so many harmonic oscillators involved that known from П/2 as the phase differences representing between the two energy lavels. We know that the above of both U(T) is the internal energy of black body radiation in thermal equilibrum. Next question what ’s the form for non thermal equilibrum? Here, I give more information, that there is a possibility to include external perturbation to the differential equation representation of the above New Planck’s Formula.

    Perhaps the New Planck’s Formula that can be useful to explain not only disclose confidential global warming but also in treating the experiment of regulated speed waves. If you and visitors here are interested to know the detail explanation about the alternative method of solving an ordinary differential equation, please visit to this address : eqworld.ipmnet.ru/forum/viewtopic.php?f=2&t=34

  9. It seems to me that the mathematical concepts developed by various individuals throughout history and all over the world today, are true or false independent of the human mind, but do not exist until discovered or communicated by those individuals.

    That’s worth giving much more thought to than I have time for now. But I’d be interested to hear one’s response to that.

  10. rohedi and Nadya Fermega, please stay on topic. Thank you.

  11. Okay sir, thank for this opportunity.

  12. Dear Mr. Robert and All,

    Before we continue this discussion there is a special request from me (maybe miss Nadya Fermega also), that I do not want to polarize the confidence in my religious beliefs with you all, especially in the divinity of each us. Let us discuss in the context of efforts to prove that mathematics is exact. Even if you include the elements of confidence in order you find out the answers why 1 +1 = 2 that is your right and I do not mind.

    Are you agree? if so, I will immediately start the discussions . Well, I will be faithful to wait your stuff.

  13. Apologize if I have been out of the discussion point by connecting the 1 +1 = 2 on the way of solving differential equation. Maybe because I do not carefully read the message in this sentence
    “how their concept of divinity plays into their ideas”.
    If so, I suggest to reveal the fact that the addition of 1 and 1, so there is no other way of thinking about what number 1 is first. For instance, we must remember that the number 1 is

    1 = 1/2 + 1/(2^2) + 1/(2^3) + 1/(2^4) + . . .

    Of course, the problem has not been finished, because there is still need
    the definition that for each real number a holds a * 1 = 1 and a * 0 = 0 where both of the relation only can be answered using miss Nadya’s question that is 1 – 1 = 0.

    Please continue the discussions.

  14. Well, I always say that math is the widest science, so if you’ve got a need you can develop systems where some laws or axioms will be different than in school mathematics. However you’ll probable find some obstacles with cohesion of your theory.
    Secondly it’ll be hard deduce something using intuition…

  15. I continue briefly brother Masteranza. Once we agree with the number 1 in the following infinity series

    1/2 + 1/(2^2) + 1/(2^3) + 1/(2^4) + … = 1

    Then we recognize the most basic operation that is multiplication that if any number multiplied by 1 will be equal to the number itself “a*1=a”. So that we can immediately write

    1 + 1/2 + 1/(2^2) + 1/(2^3) + 1/(2^4) + … = 2

    or 1 + 1 = 2 namely “1 + 1 = 2” that you want to proved. Through the exact verification above, please continue to answer all the questions you
    about how their concept of divinity plays into their ideas. Of course you will be able to prove that the opinion of Leibniz, who contributes in developing the theory of numbers.

  16. Oh yeah I forgot to itemize that

    1/2 + 1/(2^2) + 1/(2^3) + 1/(2^4) + … = 1 ,


    2*( 1/2 + 1/(2^2) + 1/(2^3) + 1/(2^4) + … ) =
    2*1 ,


    2/2 + 2* ( 1/(2^2) + 1/(2^3) + 1/(2^4) + … )=2,

    As I said before, we will know other operations of the number such as division, and the most important that the division result for a number with the number itself is always equal to 1, or “a/a=1”.

    Mr. Masteranza, I think the explanation above does not contradict with what we teach students in primary schools. So, the next task please you discuss “how the concept of divinity plays into the basic math operations”.

  17. rohedi:
    As long as I remember from set theory, proper definition for number one should be a set.

    about discuss:
    well… “divinity” is not the kind of thing which I work with and talk about, so I’ll have to stop here.

  18. You really Mr. Masteranza, I also know the proof of 1 +1 = 2 from
    Principia Mathematica book
    authorized by Alfered Nort Whitehead and Bertrand Russell on page 362. But it must be remembered that the verification is only through a preposition in which the arithmetic addition has been defined. Whereas this forum would be explored the role of divinity in the initial process of development of mathematics. If the discussion is considered finished, no problem. Many thanks for you all give the opportunity me to contribute my thoughts.

  19. Will Farris

    Leibniz is my favorite rationalist, and is simply making the elegant point that mind is different from matter. Now my favorite idealist, Berkeley, wants existence to hinge on perception, ultimately God’s perception. Ideas, such as 1+1=2, cannot exist without one’s conceiving of it, presumably. So, the idea cannot exist in a vacuum, that is, without a thinker? This is apparently an analytical proposition since the notion of “idea” entails the idea holder. If all humanity were to disappear would 1+1=2 disappear with them? Somehow, I am forceably moved to say NO! Does logic exist in the way that it does because of divine decree? Or does it exist because it cannot be otherwise, sort of like an ontological argument for the existence of abstractions apart from the abstracter (thanks, Plato). Since ideas must continue were no one to exist, and since there still must be a substratum for ideas, I posit that Dieu existe.