Abstract
The issue of the ampliation of knowledge, in particular the generation of hypotheses by ampliation of data, is not effectively treatable from a logical point of view as it is affected by the multiplicity (nay infinity) of hypotheses that can be generated by the same data. This infinity is unavoidable as a consequence of the underdetermination of data by hypotheses. The paper argues that the issue of the generation of hypotheses is, instead, treatable from a heuristic viewpoint. More specifically the paper argues that the crucial step in the generation of hypotheses is the ampliation of data, that is the integration of the data of a problem with something not contained in them. The process of ampliation is crucial in the formation of hypotheses as it narrows the infinity of possible hypotheses that explain the data. It is essentially based on ampliative inferences, in particular analogies. The paper shows that there are three main ways to ampliate data and examines and compares two case studies of generation of hypotheses. The first one is the Black-Scholes-Merton equation, namely the hypothesis that the price of an option over time is given by the partial differential equation (PDE):
The second one is the generation of the Feynman Path Integral, a hypothesis about behaviour of quantum particles (about trajectories of quantum particles), namely the hypothesis that paths followed by electrons are all possible (infinite) paths, not just the ‘classical’ ones, which can be described by the functional integral:
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Notes
- 1.
The properties, relations and entities contained in the data at the beginning of the inferential process are the ones obtained by their preliminary analysis.
- 2.
This means that it is not possible to take advantage of a price difference between two or more markets.
- 3.
Since 1926 T.B. have paid an average of \(3.8\,\%\) with very low risk both in good and bad times, while in the same interval of time the 500 stocks of S&P have paid an average of \(13\,\%\), but with great risks.
- 4.
See The Prize in Economics 1997—Press Release. Nobelprize.org. 29 Aug 2012 http://www.nobelprize.org/nobel_prizes/economics/laureates/1997/press.html.
- 5.
The assumptions explicitly used in the BSM equation are:
-
(a)
There is no arbitrage opportunity (i.e., there is no way to make a riskless profit).
-
(b)
It is possible to borrow and lend cash at a known constant risk-free interest rate.
-
(c)
It is possible to buy and sell any amount, even fractional, of stock (this includes short selling).
-
(d)
The above transactions do not incur any fees or costs (i.e., frictionless market).
-
(e)
The stock price follows a geometric Brownian motion, with constant drift and volatility.
-
(f)
The underlying security does not pay a dividend.
-
(a)
- 6.
Brownian motion is a term borrowed from physics, where it describes the motion of a molecule in a uniformly warm medium. Bachelier (see [21]) was the first to conjecture that this process can describe price changes.
- 7.
Fractal mathematics originates just from a work on cotton pricing.
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Ippoliti, E. (2014). Generation of Hypotheses by Ampliation of Data. In: Magnani, L. (eds) Model-Based Reasoning in Science and Technology. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37428-9_14
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