The Carssirer's conceptions of aprioricity, especially of synthetic a priori principles in exact sciences, is analysed. I consider his 'Marburg's' period, first of all his paper on Kant and modern mathematics. Cassirer defends the thesis on invariance principles as the modern variant of synthetic principles a priori. I analyze his arguments on the existence of apriori principles of science and compare his concept of aprioricity with holistic accounts of theories, 'semantic view of theories' and structural realism.

In this paper I discuss Mark Steiner’s view of the contribution of mathematics to physics and take up some of the questions it raises. In particular, I take up the question of discovery and explore two aspects of this question – a metaphysical aspect and a related epistemic aspect. The metaphysical aspect concerns the formal structure of the physical world. Does the physical world have mathematical or formal features or constituents, and what is the nature of these constituents? The related (...) epistemic question concerns humans’ cognitive ability to reach the formal structure of the physical world. Among other things, I explore the interaction of mathematical and non-mathematical cognition in physical discovery. (shrink)

Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum computer is not Turing machine – (...) Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 of their paper – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and psi-function – From a qualitative to a quantitative expression of relativity − psi-function, or the discrete by the random – Bartlett’s approach − psi-function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the as-sociated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Encoding a world line by psi-function – Spacetime and qubit − psi-function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = psi-function + self-adjoint operator – Reality and description – Does a „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The psi-function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocity-probability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − psi-function as physical quantity – Things in the world and instances in time – The generation of the physi-cal by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new view-point – The exception of two dimensions and Fermat’s last theorem. (shrink)

I critically examine confirmational holism as it pertains to the indispensability arguments for mathematical Platonism. I employ a distinction between pure and applied mathematics that grows out of the often overlooked symbiotic relationship between mathematics and science. I argue that this distinction undercuts the notion that mathematical theories fall under the holistic scope of the confirmation of our scientific theories.Keywords: Confirmational holism; Indispensability argument; Mathematics; Application; Science.

This article is primarily concerned with the articulation of a defensible position on the relevance of phenomenological analysis with the current epistemological edifice as this latter has evolved since the rupture with the classical scientific paradigm pointing to the Newtonian-Leibnizian tradition which took place around the beginning of 20th century. My approach is generally based on the reduction of the objects-contents of natural sciences, abstracted in the form of ideal objectivities in the corresponding logical-mathematical theories, to the content of meaning-acts (...) ultimately referring to a specific being-within-the-world experience. This is a position that finds itself in line with Husserl’s gradual departure from the psychologistic interpretations of his earlier works on the philosophy of logic and mathematics and culminates in a properly meant phenomenological foundation of natural sciences in his last major published work, namely the Crisis of European Sciences and the Transcendental Phenomenology. Further this article tries to set up a context of discourse in which to found both physical and formal objects in parallel terms as essentially temporal-noematic objects to the extent that they may be considered as invariants of the constitutional modes of a temporal consciousness. (shrink)

The paper outlines a novel version of mathematical structuralism related to invariants. The main objective here is twofold: first, to present a formal theory of structures based on the structuralist methodology underlying work with invariants. Second, to show that the resulting framework allows one to model several typical operations in modern mathematical practice: the comparison of invariants in terms of their distinctive power, the bundling of incomparable invariants to increase their collective strength, as well as a heuristic principle related to (...) the search for new invariants. (shrink)

In 1926 and 1927, James W. Alexander and Kurt Reidemeister claimed to have made “the same” crucial breakthrough in a branch of modern topology which soon thereafter was called knot theory. A detailed comparison of the techniques and objects studied in these two roughly simultaneous episodes of mathematical research shows, however, that the two mathematicians worked in quite different mathematical traditions and that they drew on related, but distinctly different epistemic resources. These traditions and resources were local, not universal elements (...) of mathematical culture. Even certain common features of the main publications such as their modernist, formal style of exposition can be explained by reference to particular constellations in the intellectual and professional environments of Alexander and Reidemeister. In order to analyze the role of such elements and constellations of mathematical research practice, a historiographical perspective is developed which emphasizes parallels with the recent historiography of experiment. In particular, a notion characterizing those “working units of scientific knowledge production” which Hans-Jörg Rheinberger has termed “experimental systems” in the case of empirical sciences proves helpful in understanding research episodes such as those bringing about modern knot theory. (shrink)

I present a theory of justification for mathematical beliefs that is both non‐foundationalist, in that it claims that some mathematics must be justified indirectly in terms of its consequences, and holistic, in that it maintains that no claim of theoretical science can be confirmed or refuted in isolation but only as a part of a system of hypotheses. Our evidence for mathematics is ultimately empirical because the mathematics that is part of theoretical science, is, in principle, revisable in light of (...) experience and confirmed by experience. Following Henry Kyburg, I develop this idea by claiming that in science we use combinations of mathematical and scientific principles to develop models that allow us to calculate values that we then compare with the data. ‘Separatists’ objections to holism revolve around the claim that holism fails to respect our intuitions about mathematics, e.g. that mathematics is clearly distinct from science and that mathematical evidence comes from proofs, rather than from experience. My response to these objections is that holism can accommodate these intuitions by appealing to pragmatic rationality that, on the one hand, underwrites the special role of mathematics and bids us to treat it as if it were a priori, and, on the other, justifies, on the grounds of simplicity and success, a local conception of evidence according to which data can confirm specific hypotheses. (shrink)

Cognitive reflection is recognized as an important skill, which is necessary for making advantageous decisions. Even though gender differences in the Cognitive Reflection test appear to be robust across multiple studies, little research has examined the source of the gender gap in performance. In Study 1, we tested the invariance of the scale across genders. In Study 2, we investigated the role of math anxiety, mathematical reasoning, and gender in CRT performance. The results attested the measurement equivalence of the Cognitive (...) Reflection Test – Long, when administered to male and female students. Additionally, the results of the mediation analysis showed an indirect effect of gender on CRT-L performance through mathematical reasoning and math anxiety. The direct effect of gender was no longer statistically significant after accounting for the other variables. The current findings suggest that cognitive reflection is affected by numerical skills and related feelings. (shrink)

It is tempting to think that a process of choosing a point at random from the surface of a sphere can be probabilistically symmetric, in the sense that any two regions of the sphere which differ by a rotation are equally likely to include the chosen point. Isaacs, Hájek, and Hawthorne (2022) argue from such symmetry principles and the mathematical paradoxes of measure to the existence of imprecise chances and the rationality of imprecise credences. Williamson (2007) has argued from a (...) related symmetry principle to the failure of probabilistic regularity. We contend that these arguments fail, because they rely on auxiliary assumptions about probability which are inconsistent with symmetry to begin with. We argue, moreover, that symmetry should be rejected in light of this inconsistency, and because it has implausible decision- theoretic implications. -/- The weaker principle of probabilistic invariance says that the probabilistic comparison of any two regions is unchanged by rotations of the sphere. This principle supports a more compelling argument for imprecise probability. We show, however, that invariance is incompatible with mundane judgments about what is probable. Ultimately, we find reason to be suspicious of the application of principles like symmetry and invariance to nonmeasurable regions. (shrink)

Although the invariance criterion of logicality first emerged as a criterion of a purely mathematical interest, it has developed into a criterion of considerable linguistic and philosophical interest. In this paper I compare two different perspectives on this criterion. The first is the perspective of natural language. Here, the invariance criterion is measured by its success in capturing our linguistic intuitions about logicality and explaining our logical behavior in natural-linguistic settings. The second perspective is more theoretical. Here, the invariance criterion (...) is used as a tool for developing a theoretical foundation of logic, focused on a critical examination, explanation, and justification of its veridicality and modal force. (shrink)

Many philosophers are baffled by necessity. Humeans, in particular, are deeply disturbed by the idea of necessary laws of nature. In this paper I offer a systematic yet down to earth explanation of necessity and laws in terms of invariance. The type of invariance I employ for this purpose generalizes an invariance used in meta-logic. The main idea is that properties and relations in general have certain degrees of invariance, and some properties/relations have a stronger degree of invariance than others. (...) The degrees of invariance of highly-invariant properties are associated with high degrees of necessity of laws governing/describing these properties, and this explains the necessity of such laws both in logic and in science. This non-mysterious explanation has rich ramifications for both fields, including the formality of logic and mathematics, the apparent conflict between the contingency of science and the necessity of its laws, the difference between logical-mathematical, physical, and biological laws/principles, the abstract character of laws, the applicability of logic and mathematics to science, scientific realism, and logical-mathematical realism. (shrink)

The indispensability argument is sometimes seen as weakened by its reliance on a controversial premise of confirmation holism. Recently, some philosophers working on the indispensability argument have developed versions of the argument which, they claim, do not rely on holism. Some of these writers even claim to have strengthened the argument by eliminating the controversial premise. I argue that the apparent removal of holism leaves a lacuna in the argument. Without the holistic premise, or some other premise (...) which facilitates the transfer of evidence to mathematical portions of scientific theories, the argument is implausible. (shrink)

A holistic account of the meaning of theoretical terms leads scientific realism into serious troubles. Alternative methods of reference fixing are needed by a realist who wishes to show how reference invariance is possible in spite of meaning variance. This paper argues that the similarity theory of truthlikeness and approximate truth, developed by logicians since the mid 1970s, helps to make precise the idea of charitable theoretical reference. Comparisons to the recent proposals by Kitcher and Psillos are given. This argument (...) helps to undermine the scepticist meta-induction about theories, and thereby to reevaluate Laudan's alleged confutation of scientific realism. (shrink)

In order to address to the relation between philosophy and mathematics it is first necessary to distinguish the grand style and the little style. The little style painstakingly constructs mathematics as the object for philosophical scrutiny. It is called the little style for a precise reason, because it assigns mathematics to the subservient role of that which supports the definition and perpetuation of a philosophical specialisation. This specialisation is called the ‘philosophy of mathematics’, where the ‘of’ is objective. The philosophy (...) of mathematics can in turn be inscribed under the area of specialisation that supports the name ‘epistemology and history of science’, an area to which corresponds a specialised bureaucracy in the academic authorities and committees whose role it is to manage the personnel of researchers and teachers. But in philosophy, specialisation is invariably the means by which the little style insinuates itself. In Lacan terms, this occurs through the collapse of the discourse of the Master, which is rooted in the signifier of the same name, the S1 that gives rise to a signifying chain, onto the discourse of the University, that perpetual commentary which adequately represents the second moment of all speech, that is, the S2 which only exists by making the Master disappear under the commentary which exhausts it. The little style of the philosophy of mathematics, and of its epistemology, strives for such a disappearance of the ontological sovereignty of mathematics, its instituting aristocratism, its unrivalled mastery, by confining its dramatic and almost incomprehensible existence to a generally dusty compartment of academic specialisation. (shrink)

It is now standard to interpret symmetry-related models of physical theories as representing the same state of affairs. Recently, a debate has sprung up around the question when this interpretational move is warranted. In particular, Møller-Nielsen :1253–1264, 2017) has argued that one is only allowed to interpret symmetry-related models as physically equivalent when one has a characterisation of their common content. I disambiguate two versions of this claim. On the first, a perspicuous interpretation is required: an account of the models’ (...) common ontology. On the second, stricter, version of this claim, a perspicuous formalism is required in addition: one whose mathematical structures ‘intrinsically’ represent the physical world, in the sense of Field. Using Dewar’s :485–521, 2019) distinction between internal and external sophistication as a case study, I argue that the second requirement is decisive. This clarifies the conditions under which it is warranted to interpret symmetry-related models as physically equivalent. (shrink)

One questioned premiss in the indispensability argument of Quine and Putnam is confirmational holism. In this paper I argue for a weakened form of holism, and thus a strengthened version of the ind ..

Properties and relations in general have a certain degree of invariance, and some types of properties/relations have a stronger degree of invariance than others. In this paper I will show how the degrees of invariance of different types of properties are associated with, and explain, the modal force of the laws governing them. This explains differences in the modal force of laws/principles of different disciplines, starting with logic and mathematics and proceeding to physics and biology.

The isomorphism invariance criterion of logical nature has much to commend it. It can be philosophically motivated by the thought that logic is distinctively general or topic neutral. It is capable of precise set-theoretic formulation. And it delivers an extension of ‘logical constant’ which respects the intuitively clear cases. Despite its attractions, the criterion has recently come under attack. Critics such as Feferman, MacFarlane and Bonnay argue that the criterion overgenerates by incorrectly judging mathematical notions as logical. We consider five (...) possible precisifications of the overgeneration argument and find them all unconvincing. (shrink)

This volume is the proceedings of a workshop to discuss the recent work on complex systems in physics and biology, its epistemological and cultural implications, and its effect for the development of these two sciences. The workshop is geared towards physicists, biologists, and science historians.

This seminal, multidisciplinary book shows how mathematics can be used to study the first principles of DNA. Most importantly, it enriches the so-called "Chargaff's grammar of biology" by providing the conceptual theoretical framework necessary to generalize Chargaff's rules. Starting with a simple example of DNA mathematical modeling where human nucleotide frequencies are associated to the Fibonacci sequence and the Golden Ratio through an optimization problem, its breakthrough is showing that the reverse, complement and reverse-complement operators defined over oligonucleotides induce a (...) natural set partition of DNA words of fixed-size. These equivalence classes, when organized into a matrix form, reveal hidden patterns within the DNA sequence of every living organism. Intended for undergraduate and graduate students both in mathematics and in life sciences, it is also a valuable resource for researchers interested in studying invariant genomic properties. (shrink)

The indispensability argument is a method for showing that abstract mathematical objects exist. Various versions of this argument have been proposed. Lately, commentators seem to have agreed that a holistic indispensability argument will not work, and that an explanatory indispensability argument is the best candidate. In this paper I argue that the dominant reasons for rejecting the holistic indispensability argument are mistaken. This is largely due to an overestimation of the consequences that follow from evidential holism. Nevertheless, the holistic (...) indispensability argument should be rejected, but for a different reason —in order that an indispensability argument relying on holism can work, it must invoke an unmotivated version of evidential holism. Such an argument will be unsound. Correcting the argument with a proper construal of evidential holism means that it can no longer deliver mathematical Platonism as a conclusion: such an argument for Platonism will be invalid. I then show how the reasons for rejecting the holistic indispensability argument importantly constrain what kind of account of explanation will be permissible in explanatory versions. (shrink)

The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to homotopy (...) type theory. It also gives the new system of foundations a distinctly structural character. (shrink)

This paper argues for confirmational holism about facts and values. This position is similar to one defended by (among others) Hilary Putnam, but the argument is importantly different. Whereas Putnam et al. rely on examples of the putative entanglement of facts and values – a strategy which I suggest is vulnerable to parrying – my argument proceeds at a more general level. I argue that the explanation of action can not be separated from our practical reasoning, and for this (...) reason, the ‘webs’ of value and fact judgments are joined in the same way that Quine holds the judgments of mathematics and natural science to be. (shrink)

In this note we outline the history of q-deformations, indicate their physical shortcomings, suggest their apparent resolution via an invariant Lie-admissible formulation based on a new mathematics of genotopic type, and point out their expected physical significance.

Idealization is commonly understood as distortion: representing things differently than how they actually are. In this paper, we outline an alternative artifactual approach that does not make misrepresentation central for the analysis of idealization. We examine the contrast between the Hodgkin-Huxley (1952a, b, c) and the Heimburg-Jackson (2005, 2006) models of the nerve impulse from the artifactual perspective, and argue that, since the two models draw upon different epistemic resources and research programs, it is often difficult to tell which features (...) of a system the central assumptions involved are supposed to distort. Many idealizations are holistic in nature. They cannot be locally undone without dismantling the model, as they occupy a central position in the entire research program. Nor is their holistic character mainly related to the use of mathematical and statistical modeling techniques as portrayed by Rice (2018, 2019). We suggest that holistic idealizations are implicit theoretical and representational assumptions that can only be understood in relation to the conceptual and representational tools exploited in modeling and experimental practices. Such holistic idealizations play a pivotal role not just in individual models, but also in defining research programs. -/- . (shrink)

Revision sequences are a kind of transfinite sequences which were introduced by Herzberger and Gupta in 1982 as the main mathematical tool for developing their respective revision theories of truth. We generalise revision sequences to the notion of cofinally invariant sequences, showing that several known facts about Herzberger’s and Gupta’s theories also hold for this more abstract kind of sequences and providing new and more informative proofs of the old results.

Mathematics as a Science of Patterns is the definitive exposition of a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defence of realism (...) about the metaphysics of mathematics--the view that mathematics is about things that really exist. Resnik's distinctive philosophy of mathematics is here presented in an accessible and systematic form: it will be of value not only to specialists in this area, but to philosophers, mathematicians, and logicians interested in the relationship between these three disciplines, or in truth, realism, and epistemology. (shrink)

A moda logic Λ is called invariant if for all automorphisms α of NExt K, α = Λ. An invariant ogic is therefore unique y determined by its surrounding in the attice. It wi be established among other that a extensions of K.alt1S4.3 and G.3 are invariant ogics. Apart from the results that are being obtained, this work contributes to the understanding of the combinatorics of finite frames in genera, something wich has not been done except for transitive frames. Certain (...) useful concepts will be established, such as the notion of a d-homogeneous frame. (shrink)

The failed criterion of logical truth proposed by Carnap in the Logical Syntax of Language was based on the determinateness of all logical and mathematical statements. It is related to a conception which is independent of the specifics of the system of the Syntax, hints of which occur elsewhere in Carnap’s writings, and those of others. What is essential is the idea that the logical terms are invariant under reinterpretation of the empirical terms, and are therefore semantically determinate. A certain (...) objection to Carnap’s version of the invariance conception has been repeated several times in the literature. It is based on Gödel incompleteness, which is puzzling, since Carnap’s Syntax is otherwise quite careful to take account of Gödel. We show here that, in fact, the objection is invalid and is based on a confusion about determinacy. Sorting this out is worthwhile not only for the purpose of better understanding Carnap’s thinking in the Syntax, though. The invariance conception is also related to recent work in the philosophy of logic regarding “logicality”—the characterization of logical concepts—following a proposal of Tarski’s. It is even connected to some very recent developments in the foundations of mathematics. (shrink)

The Quine-Putnam Indispensability argument is the argument for treating mathematical entities on a par with other theoretical entities of our best scientific theories. This argument is usually taken to be an argument for mathematical realism. In this chapter I will argue that the proper way to understand this argument is as putting pressure on the viability of the marriage of scientific realism and mathematical nominalism. Although such a marriage is a popular option amongst philosophers of science and mathematics, in light (...) of the indispensability argument, the marriage is seen to be very unstable. Unless one is careful about how the Quine-Putnam argument is disarmed, one can be forced to either mathematical realism or, alternatively, scientific instrumentalism. I will explore the various options: (i) finding a way to reconcile the two partners in the marriage by disarming the indispensability argument (Jody Azzouni [2], Hartry Field [13, 14], Alan Musgrave [18, 19], David Papineau [21]); (ii) embracing mathematical realism (W.V.O. Quine [23], Michael Resnik [25], J.J.C. Smart [27]); and (iii) embracing some form of scientific instrumentalism (Ot´ avio Bueno [7, 8], Bas van Fraassen [30]). Elsewhere [11], I have argued for option (ii) and I won’t repeat those arguments here. Instead, I will consider the difficulties for each of the three options just mentioned, with special attention to option (i). In relation to the latter, I will discuss an argument due to Alan Musgrave [19] for why option (i) is a plausible and promising approach. From the discussion of Musgrave’s argument, it will emerge that the issue of holist versus separatist theories of confirmation plays a curious role in the realism–antirealism debate in the philosophy of mathematics. I will argue that if you take confirmation to be an holistic matter—it’s whole theories (or significant parts thereof) that are confirmed in any experiment—then there’s an inclination to opt for (ii) in order to resolve the marital tension outlined above.. (shrink)

We study invariant types in NIP theories. Amongst other things: we prove a definable version of the [Formula: see text]-theorem in theories of small or medium directionality; we construct a canonical retraction from the space of [Formula: see text]-invariant types to that of [Formula: see text]-finitely satisfiable types; we show some amalgamation results for invariant types and list a number of open questions.

The paper presents two case studies of multi-agent information exchange involving generalized quantifiers. We focus on scenarios in which agents successfully converge to knowledge on the basis of the information about the knowledge of others, so-called Muddy Children puzzle and Top Hat puzzle. We investigate the relationship between certain invariance properties of quantifiers and the successful convergence to knowledge in such situations. We generalize the scenarios to account for public announcements with arbitrary quantifiers. We show that the Muddy Children puzzle (...) is solvable for any number of agents if and only if the quantifier in the announcement is positively active (satisfies a version of the variety condition). In order to get the characterization result, we propose a new concise logical modeling of the puzzle based on the number triangle representation of generalized quantifiers. In a similar vein, we also study the Top Hat puzzle. We observe that in this case an announcement needs to satisfy stronger conditions in order to guarantee solvability. Hence, we introduce a new property, called bounded thickness, and show that the solvability of the Top Hat puzzle for arbitrary number of agents is equivalent to the announcement being 1-thick. (shrink)

A topological inevitability of early developmental events through the use of classical topological concepts is discussed. Topological dynamics of forms and maps in embryo development are presented. Forms of a developing organism such as cell sets and closed surfaces are topological objects. Maps (or mathematical functions) are additional topological constructions in these objects and include polarization, singularities and curvature. Topological visualization allows us to analyze relationships that link local morphogenetic processes and integral developmental structures and also to find stable spatio-temporal (...) topological characteristics that are invariant for a taxonomic group. The application of topological principles reveals a topological imperative: certain topological rules define and direct embryogenesis. A topological stability of embryonic morphogenesis is proposed and a topological scenario of developmental and evolutionary transformations is presented. (shrink)

For the probabilistic description of all the joint von Neumann measurements on a D-dimensional quantum system, we present the specific example of a context-invariant quasi hidden variable model, proved in Loubenets to exist for each Hilbert space. In this model, a quantum observable X is represented by a variety of random variables satisfying the functional condition required in quantum foundations but, in contrast to a contextual model, each of these random variables equivalently models X under all joint von Neumann measurements, (...) regardless of their contexts. This, in particular, implies the specific local qHV model for an N-qudit state and allows us to derive the new exact upper bound on the maximal violation of 2\2-setting Bell-type inequalities of any type under N-partite joint von Neumann measurements on an N-qudit state. For d = 2, this new upper bound coincides with the maximal violation by an N-qubit state of the Mermin–Klyshko inequality. Based on our results, we discuss the conceptual and mathematical advantages of context-invariant and local qHV modelling. (shrink)

It is claimed that the indispensability argument for the existence of mathematical entities (IA) works in a way that allows a proponent of mathematical realism to remain agnostic with regard to how we establish that mathematical entities exist. This is supposed to be possible by virtue of the appeal to confirmational holism that enters into the formulation of IA. Holism about confirmation is supposed to be motivated in analogy with holism about falsification. I present an account of (...) how holism about falsification is supposed to be motivated. I argue that the argument for holism about falsification is in tension with how we think about confirmation and with two principles suggested by Quine for construing a plausible variety of holism. Finally, I show that one of Quine's principles does not allow a proponent of mathematical realism to remain agnostic with regard to how we establish that mathematical entities exist. (shrink)

This paper argues for confirmational holism about facts and values. This position is similar to one defended by (among others) Hilary Putnam, but the argument is importantly different. Whereas Putnam et al. rely on examples of the putative entanglement of facts and values – a strategy which I suggest is vulnerable to parrying – my argument proceeds at a more general level. I argue that the explanation of action can not be separated from our practical reasoning, and for this (...) reason, the ‘webs’ of value and fact judgments are joined in the same way that Quine holds the judgments of mathematics and natural science to be. (shrink)

The prevalent interpretation of Gödel’s Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of the underlying mathematical theorem depends on several arbitrary formalisation choices. In this paper I examine the theorem’s dependency regarding Gödel numberings. I introducedeviantnumberings, yielding provability predicates satisfying Löb’s conditions, which result in provable consistency sentences. According to the main result of this paper however, these (...) “counterexamples” do not refute the theorem’s prevalent interpretation, since once a natural class ofadmissiblenumberings is singled out, invariance is maintained. (shrink)

"Intuition" has perhaps been the least understood and the most abused term in philosophy. It is often the term used when one has no plausible explanation for the source of a given belief or opinion. According to some sceptics, it is understood only in terms of what it is not, and it is not any of the better understood means for acquiring knowledge. In mathematics the term has also unfortunately been used in this way. Thus, intuition is sometimes portrayed as (...) if it were the Third Eye, something only mathematical "mystics", like Ramanujan, possess. In mathematics the notion has also been used in a host of other senses: by "intuitive" one might mean informal, or non-rigourous, or visual, or holistic, or incomplete, or perhaps even convincing in spite of lack of proof. My aim in this book is to sweep all of this aside, to argue that there is a perfectly coherent, philosophically respectable notion of mathematical intuition according to which intuition is a condition necessary for mathemati cal knowledge. I shall argue that mathematical intuition is not any special or mysterious kind of faculty, and that it is possible to make progress in the philosophical analysis of this notion. This kind of undertaking has a precedent in the philosophy of Kant. While I shall be mostly developing ideas about intuition due to Edmund Husser! there will be a kind of Kantian argument underlying the entire book. (shrink)

Not all symmetries are on a par. For instance, within Newtonian mechanics, we seem to have a good grasp on the empirical significance of boosts, by applying it to subsystems. This is exemplified by the thought experiment known as Galileo’s ship: the inertial state of motion of a ship is immaterial to how events unfold in the cabin, but is registered in the values of relational quantities such as the distance and velocity of the ship relative to the shore. But (...) the significance of gauge symmetries seems less clear. For example, can gauge transformations in Yang-Mills theory—taken as mere descriptive redundancy—exhibit a similar relational empirical significance as the boosts of Galileo’s ship? This question has been debated in the last fifteen years in philosophy of physics. I will argue that the answer is ‘yes’, but only for a finite subset of gauge transformations, and under special conditions. Under those conditions, we can mathematically identify empirical significance with a failure of supervenience: the state of the Universe is not uniquely determined by the intrinsic state of its isolated subsystems. Empirical significance is therefore encoded in those relations between subsystems that stand apart from their intrinsic states. (shrink)