Two Criticisms against Mathematical Realism

Mathematical realism asserts that mathematical objects exist in the abstract world, and that a mathematical sentence is true or false, depending on whether the abstract world is as the mathematical sentence says it is. I raise two objections against mathematical realism. First, the abstract world is queer in that it allows for contradictory states of affairs. Second, mathematical realism does not have a theoretical resource to explain why a sentence about a tricle is true or false. A tricle is an object that changes its shape from a triangle to a circle, and then back to a triangle with every second

Can we have mathematical understanding of physical phenomena?

Can mathematics contribute to our understanding of physical phenomena? One way to try to answer this question is by getting involved in the recent philosophical dispute about the existence of mathematical explanations of physical phenomena. If there is such a thing, given the relation between explanation and understanding, we can say that there is an affirmative answer to our question. But what if we do not agree that mathematics can play an explanatory role in science? Can we still consider that the above question can have an affirmative answer? My main aim here is to give an account that takes mathematics, in some of the cases discussed in the literature, as contributing to our understanding of physical phenomena despite not being explanatory

The Enhanced Indispensability Argument, the circularity problem, and the interpretability strategy

Within the context of the Quine-Putnam indispensability argument, one discussion about the status of mathematics is concerned with the `Enhanced Indispensability Argument', which makes explicit in what way mathematics is supposed to be indispensable in science, namely explanatory. If there are genuine mathematical explanations of empirical phenomena, an argument for mathematical platonism could be extracted by using inference to the best explanation. The best explanation of the primeness of the life cycles of Periodical Cicadas is genuinely mathematical, according to Baker (2005, 2009). Furthermore, the result is then also used to strengthen the platonist position (e.g. Baker 2017a). We pick up the circularity problem brought up by Leng (2005) and Bangu (2008). We will argue that Baker's attempt to solve this problem fails, if Hume's Principle is analytic. We will also provide the opponent of the Enhanced Indispensability Argument with the so-called 'interpretability strategy', which can be used to come up with alternative explanations in case Hume's Principle is non-analytic. \end{abstract

Why inference to the best explanation doesn’t secure empirical grounds for mathematical platonism

Proponents of the explanatory indispensability argument for mathematical platonism maintain that claims about mathematical entities play an essential explanatory role in some of our best scientific explanations. They infer that the existence of mathematical entities is supported by way of inference to the best explanation from empirical phenomena and therefore that there are the same sort of empirical grounds for believing in mathematical entities as there are for believing in concrete unobservables such as quarks. I object that this inference depends on a false view of how abductive considerations mediate the transfer of empirical support. More specifically, I argue that even if inference to the best explanation is cogent, and claims about mathematical entities play an essential explanatory role in some of our best scientific explanations, it doesn’t follow that the empirical phenomena that license those explanations also provide empirical support for the claim that mathematical entities exist

Can Arguments of Formal Naturalism be used to Show that the Mathematical Explanation is Indispensable in Science?

Poznato je da platonisti u filozofiji matematike podržavaju gledište o postojanju matematičkih objekata. Takozvani Pojačani argument neizostavnosti (Enhanced Indispensability Argument EIA) koji je Alan Baker nedavno eksplicitno formulirao u obliku modalnog silogizma, može se shvatiti kao pokušaj da se za ovo platonističko stanovište nađe jedan oslonac. U skorije vrijeme ovaj argument izazvao je velik broj različitih reakcija. Manji broj analiza podržavao je Argument ili neki njegov dio. Mi ćemo izdvojiti upravo jednu takvu analizu kojom je podržana druga premisa EIA. Riječ je o vrsti naturalističkog pristupa pitanjima uloge i neizostavnosti matematičkog objašnjenja u znanosti. Nastojat ćemo pokazati da ovaj pokušaj obrane spomenute premise, a time ujedno i EIA, ima nekoliko značajnih nedostataka zbog kojih nam se čini da ne može biti dodatan oslonac za platoniste.In the philosophy of mathematics, it is well known that the Platonists support the view of the existence of mathematical objects. The so-called Enhanced indispensability argument EIA, recently explicitly formulated by Alan Baker in the form of modal syllogisms, can be understood as an attempt to support this Platonic view. this argument has recently caused a unmber of different reactions. A small number of analyses supported the argument or any of its parts. We will single out exactly one such analysis which supports the second premise of the EIA. It is a naturalistic approach to the role and indispensability of mathematical explanation in science. We will try to show that this attempt to defend the said premise and hence the EIA has several significant shortcomings due to which, it seems to us, it cannot serve as an additional argument in favour of the Platonists

Mathematical Explanation and Ontology: An Analysis of Applied Mathematics and Mathematical Proofs

The present work aims at providing an account of mathematical explanation in two different areas: scientific explanation and within mathematics. The research is addressed from two different perspectives: the one arising from an ontological concern about mathematical entities, and the other originating from a methodological choice: to study our chosen problems (mathematical explanation in science and in mathematics itself) in mathematical practice, that is to say, looking at the way mathematicians understand and perform their work in these diverse areas, including a case study for the context of intra-mathematical explanation. The central target is the analysis of the role that mathematical explanation plays in science and its relevance to the success or failure of scientific theories. The ontological question of whether the explanatory role of abstract objects, mathematical objects in particular, is enough to postulate their existence will be one of the issues to be addressed. Moreover, the possibility of a unified theory of explanation which can accommodate both external and internal mathematical explanation will also be considered. In order to go deeper into these issues, the research includes: (1) an analysis how the question of what is involved in internal mathematical explanation has been addressed in the literature, an analysis of the role of mathematical proof and the reasons why it makes sense to search for more explanatory proofs of already known results, and (2) an analysis of the relation between the use of mathematics in scientific explanation and the ontological commitment that arises from these explanatory tools in science. Part of the present work consists of an analysis of the explanatory role of mathematics through the study of cases reflecting this role. Case studies is one of the main sources of data in order to clarify the role mathematical entities play, among other methodological resources