Math and science were the last border guards standing on the premises of what could be analyzed by sociologists, specifically because technoscience was rooted in reality (Collins, 2014). We take mathematics for granted, as one of the immutable pillars of truth upon which we build technoscience; yet math cannot be detached from the sociohistorical context of the mathematician as well as power relations within that context (Bauchspies et al, 2015).
While in an academic environment we’re encouraged to discuss and question our assumptions, it is uncommon for non-math graduates to question the nature of the axioms that rule our experiments. What happens when we actually do it? What happens when we dare to question our own assumptions?
I guess we would all agree that the more numbers are added to the sum, the more the result will grow:
[1] 1+2+3+… = ∞
It makes complete sense; such a statement is universally accepted, empirically validated and actively sought after. It’s so deeply rooted within our belief system that almost no one dares to question it. The act of questioning one’s own beliefs might result into an unexpected journey, one of such examples can be found through some of the equations among those proposed by Srinivasa Ramanujan:
[2] 1-1+1-1+…=1/2
[3] 1-2+3-4+… = 1/4
[4] 1+2+3+4+… = – 1/12
When looking at equations [2], [3] and [4], someone doesn’t quite add up. For equation [2], we have either 1 or 0, how does that ½ come to existence? For equation [3] there is also no hint that the sum will converge to a certain number. And well… equation [4] is just special, it is the equivalent of being a blind man in a dark room, looking for a black cat that isn’t there. It states that the addition of positive numbers actually leads to a constant, but no any constant, a negative one! While equation [4] is certainly the most fun, it would take me more than a blog post to explain.
At this point of the journey, the itch on our backs derives from the fact of trying to make sense out of equations [2] and [3]. Both of them state that even as the numbers continue for unlimited sums, there will always be a discrete answer. The equations converge. But what is convergence?
According to what most of would agree, convergence would be represented as in equation [5]
In which is the n-th partial sum. The sum goes from k=1 to whichever number we want to set for n. And represents the sequence we want to analyze. In plain terms, equation [5] allows us to determine Sn at any given time. When there is a number we can roughly define for , then convergence has been achieved.
When looking back at equation [2], the partial sums are either 1 or 0, alternating the whole time without a stable number to choose. For equation [3] the partial sums keep alternating in sign and increasing in number, so there is also no stable . The only possible option becomes then to question the definition of convergence itself. And more importantly to question our own notion of convergence as an almighty definition and to accept other definitions.
This question has been around for a while and that’s why Cesàro convergence (See equation [6]) came to existence. Such a definition follows the same logic of the definition used in equation [5], yet there is one crucial difference: For Cesàro, convergence is achieved through the means.
While such an expression might look like the fuel for nightmares, it simply means that the if the arithmetic mean (defined as ) of the first n partial sums tends to A, then it converges. This newly acquired definition allows us to look through a different lens at equations [2] & [3]
When looking back at equation [2], we know that the partial sums revolve around 1 and 0. If we take the mean out of both numbers, then we arrive at Ramanujan’s ½. The same applies for equation [3], yet in this case we need to calculate the means of each partial sum and then the means out of those means, kind of a “meansception”.
When questioning convergence we came with up to three different explanations, either the “normal” definition, Cesáro or Cesáro reloaded. The question then becomes, which one of those is correct? Well, all of them are correct, they just point at different axioms upon which math can be built.
When returning to the question whether math is within the boundaries of sociology, it would be easy to state that both of Cesàro approaches don’t make sense in our world. We know for a fact that the more is added, the bigger the result. Yet the harmonic series and the Riemann Zeta function can be explained through Cesàro convergence. Furthermore, Cesàro’s definition can explain “normal” convergence, whereas (and as we experienced above), this won’t function the other way around. Additionally, if we go by Popper’s principle of falsification, neither approach can be falsified. When dealing with abstractness at such a level, we incur into the blurry borders between math and philosophy.
Then why isn’t Cesàro the standard definition for convergence? Or more importantly, how different would be our worldview if when thinking about infinity, Cesàro’s convergence would come up to our heads rather than the “standard” definition? Or more importantly, how many of math’s axioms have a better explanation, yet we stick to them out of habit? in terms of Plato’s Theory of Forms, is our mathematical understanding only a parameter of a wider more comprehensive concept of math?
For me, the most important question is: how many theories and axioms, which provide better mathematical explanation, are buried behind the politics and egos of mathematicians who created the paradigms upon we live now?
References:
Bauchspies, W. K., Croissant, J., & Restivo, S. (2006). Science, Technology and Society – A Sociological Approach.
Berndt, B. C. (2012). Ramanujan’s notebooks. Springer Science & Business Media.
Cesàro summation methods. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Ces%C3%A0ro_summation_methods&oldid=26199
Collins, H. M. (2014). Are we all scientific experts now? Cambridge: Polity.
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