Towards a Theory of Mathematical Argument

In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument.     

Systems of Logical Systems: Neuroscience and Quantum Logic

Nervous systems are intricately organized on many levels of analysis.The intricate organization invites the development of mathematicalsystems that reflect its logical structure. Particular logical structures and choices of invariants within those structures narrowthe ranges of perceptions that are possible and sensorimotorcoordination that may be selected. As in quantum logic, choicesaffect outcomes. Some of the mathematical tools in use in quantum logic havealready also been used in neurobiology, including the mathematicsof ordered structures and a product like a tensor product. Astheoretical neurobiology is developed on its own foundation, wemay expect a rich dialogue between theoretical neurobiology andquantum logic.     

表征还是建构? 量子场论的一种解释

在本文中,我坚持认为,量子场论所描述的具有因果关系的实体层次中那些基本的实体不是粒子而是场。然后,我从结构实在论的角度进一步讨论了在何种意义上并在何种程度上,有关场的这种理论建构可以被看作物理实在的客观表征。     

Quantum Particles as Conceptual Entities: A Possible Explanatory Framework for Quantum Theory

We put forward a possible new interpretation and explanatory framework for quantum theory. The basic hypothesis underlying this new framework is that quantum particles are conceptual entities. More concretely, we propose that quantum particles interact with ordinary matter, nuclei, atoms, molecules, macroscopic material entities, measuring apparatuses,  in a similar way to how human concepts interact with memory structures, human minds or artificial memories. We analyze the most characteristic aspects of quantum theory, i.e. entanglement and non-locality, interference and superposition, identity and individuality in the light of this new interpretation, and we put forward a specific explanation and understanding of these aspects. The basic hypothesis of our framework gives rise in a natural way to a Heisenberg uncertainty principle which introduces an understanding of the general situation of ‘the one and the many’ in quantum physics. A specific view on macro and micro different from the common one follows from the basic hypothesis and leads to an analysis of Schrödinger’s Cat paradox and the measurement problem different from the existing ones. We reflect about the influence of this new quantum interpretation and explanatory framework on the global nature and evolutionary aspects of the world and human worldviews, and point out potential explanations for specific situations, such as the generation problem in particle physics, the confinement of quarks and the existence of dark matter.     

A Further Review of the Incompatibility between Classical Principles and Quantum Postulates

The traditional “realist” conception of physics, according to which human concepts, laws and theories can grasp the essence of a reality in our absence, seems incompatible with quantum formalism and it most fruitful interpretation. The proof rests on the violation by quantum mechanical formalism of some fundamental principles of the classical ontology. We discuss if the conception behind Einstein’s idea of a reality in our absence, could be still maintained and at which price. We conclude that quantum mechanical formalism is not formulated on those terms, leaving for a separated paper the discussion about the terms in which it could be formulated and the onto-epistemological implications it might have.     

The Observer Effect

Founding our analysis on the Geneva-Brussels approach to the foundations of physics, we provide a clarification and classification of the key concept of observation. An entity can be observed with or without a scope. In the second case, the observation is a purely non-invasive discovery process; in the first case, it is a purely invasive process, which can involve either creation or destruction aspects. An entity can also be observed with or without a full control over the observational process. In the latter case, the observation can be described by a symmetry breaking mechanism, through which a specific deterministic observational process is selected among a number of potential ones, as explained in Aerts’ hidden measurement approach. This is what is called a product test, or product observation, whose consequences are that outcomes can only be predicted in probabilistic terms, as it is the case in typical quantum measurements. We also show that observations can be about intrinsic (stable) properties of the observed entity, or about relational (ephemeral) properties between the observer and observed entities; also, they can be about intermediate properties, neither purely classical, nor purely quantum. Our analysis allows us to propose a general conceptual characterization of quantum measurements, as observational processes involving three aspects: (1) product observations, (2) pure creation aspects and (3) ephemeral relational properties. We also discuss the important concept of non-spatiality and emphasize some of the differences and similarities between quantum and classical/relativistic observations.     

Quantum Theory and the Nature of Consciousness

Our interest focusses on the idea, that consciousness is a powerful acting entity. Up to now there does not exist a scientific concept for this idea. This is not due to problems within the field of psychology or brain research, but rather in resisting theories of modern physics. That is, why we have to search for a solution in the field of physics. A solution can be found in a new understanding of the basics of physical theory. That could be given by abstract and absolute quantum bits of information (AQI bits). To avoid the popular misunderstanding of “information” as “meaningful” it was necessary to find a new word for the free-of-meaning AQI bits: the AQI bits establish a quantum pre-structure termed “Protyposis” (Greek: “pre-formation”), out of which real objects can be formed, starting from energetical and material elementary particles. The Protyposis AQI bits provide a pre-structure for all entities in natural sciences. They are the basic entities, whereof the physical nature of the brain, on the one hand, and the mental nature of consciousness, on the other hand, were formed during the cosmological and the following biological evolution. A deeper understanding of quantum structures may help to overcome the resistance against quantum theory in the field of brain research and consciousness. The key for an understanding is the concept of Protyposis, which means an abstract quantum information free of any definite meaning. With the AQI bits of the Protyposis, both, massless and massive quantum particles can be constructed. Even quantum information with special meanings, in example grammatically formulated thoughts, eventually could be explained. As long as the fundamental basis of quantum theory is misunderstood as being formed by a manifold of some small objects like atoms, quarks, or strings, the problem of understanding consciousness has no solution. If instead we understand quantum theory as based on truly simple quantum structures, there would be no longer fundamental problems for an understanding of consciousness.     

量子逻辑解释优于哥本哈根解释吗——围绕投影法则的争论

弗里德曼和普特南认为,哥本哈根解释只能特设性地引入投影法则来符合实验事实,然而由量子逻辑则可以推导出投影法则,因此量子逻辑解释比哥本哈根解释更加优越.这一论断提出后不久就遭到了许多学者的反驳.海尔曼认为由量子逻辑推导投影法则的过程也是特设性的,巴布则认为由哥本哈根解释也可以推导出投影法则.随后,斯戴尔斯提出了由哥本哈根...     

The Intentionality of Formal Systems

One of the most interesting and entertaining philosophical discussions of the last few decades is the discussion between Daniel Dennett and John Searle on the existence of intrinsic intentionality. Dennett denies the existence of phenomena with intrinsic intentionality. Searle, however, is convinced that some mental phenomena exhibit intrinsic intentionality. According to me, this discussion has been obscured by some serious misunderstandings with regard to the concept ‘intrinsic intentionality’. For instance, most philosophers fail to realize that it is possible that the intentionality of a phenomenon is partly intrinsic and partly observer relative. Moreover, many philosophers are mixing up the concepts ‘original intentionality’ and ‘intrinsic intentionality’. In fact, there is, in the philosophical literature, no strict and unambiguous definition of the concept ‘intrinsic intentionality’. In this article, I will try to remedy this. I will also try to give strict and unambiguous definitions of the concepts ‘observer relative intentionality’, ‘original intentionality’, and ‘derived intentionality’. These definitions will be used for an examination of the intentionality of formal mathematical systems. In conclusion, I will make a comparison between the (intrinsic) intentionality of formal mathematical systems on the one hand, and the (intrinsic) intentionality of human beings on the other hand.     

Embedding Quantum Mechanics into a Broader Noncontextual Theory

Scholars concerned with the foundations of quantum mechanics (QM) usually think that contextuality (hence nonobjectivity of physical properties, which implies numerous problems and paradoxes) is an unavoidable feature of QM which directly follows from the mathematical apparatus of QM. Based on some previous papers on this issue, we criticize this view and supply a new informal presentation of the extended semantic realism (ESR) model which embodies the formalism of QM into a broader mathematical formalism and reinterprets quantum probabilities as conditional on detection rather than absolute. Because of this reinterpretation a hidden variables theory can be constructed which justifies the assumptions introduced in the ESR model and proves its objectivity. When applied to special cases the ESR model settles long-standing conflicts (it reconciles Bell’s inequalities with QM), provides a general framework in which previous results obtained by other authors (as local interpretations of the GHZ experiment) are recovered and explained, and supports an interpretation of quantum logic which avoids the introduction of the problematic notion of quantum truth.     

从逻辑哲学观点看量子逻辑

1.根据逻辑哲学,量子逻辑可以定位于一种以非分配性为特征的非经典逻辑,它以量子力学家实际使用的科学推理为现实原型或经验基础;2.普特南从量子逻辑的产生而得出根据经验的理由,逻辑是整体可修正的结论。但他同时又主张逻辑联词的意义不变,因此陷入二难困境。3.根据量子力学几种不同解释来重新审视经验理由和逻辑可修正性论证,发现形而上学考量更不可忽视。4.对应原理具有消解普特南二难的方法论功效,量子逻辑对于经典逻辑的继承性与革新性同时可以得到合理解释。     

The Future of Mathematics in Economics: A Philosophically Grounded Proposal

The use of mathematics in economics has been widely discussed. The philosophical discussion on what mathematics is remains unsettled on why it can be applied to the study of the real world. We propose to get back to some philosophical conceptions that lead to a language-like role for the mathematical analysis of economic phenomena and present some problems of interest that can be better examined in this light. Category theory provides the appropriate tools for these analytical approach.     

清末黄宗宪的《容圆七术》初探

清末数学家黄宗宪在<容圆七术>中,对中国传统数学的容圆问题从几个方面做了推广,特别是用西方数学的圆锥曲线、轨迹方法及逻辑推理等知识解决新的容圆问题,其中不乏颇具新意的创见和成果.我们由此也可看到我国清末东、西方数学交融时期的许多现象.     

量子场论的还原性问题

描述粒子物理学的核心语言是可重整化的量子场论。重整化最初是应对量子场论中数学计算无穷大问题的有效策略,是解决理论物理学中“突现”问题的有力手段。本文从重整化操作在量子场论中产生的影响出发,回顾了有效场论思想产生的历史,叙述了重整化方法所引起的哲学争论,旨在对还原论和反还原论在解释量子场论中涉及的“还原”、“突现”、“层次性”等问题予以方法论考察,最后从语境的视角对重整化方法进行诠释,指出对基础理论及理论间关系的探讨是具有境遇性的。     

我们能够获得那些可能性或必然性的知识吗?--以威廉姆森的观点为契机

模态知识论的核心问题是:"我们如何证成或知道那些仅仅属于可能性(mere possibility)的宣称,与那些既非逻辑亦非数学的必然性宣称?"对于这个问题,威廉姆森论证并主张:由于每一类的模态宣称都逻辑上等价于某一类的反事实条件句宣称,因而日常透过"想象模拟"去评估反事实条件句的认知能力也就给了我们认知形而上学模态性的能力。威廉姆森并且认为,有些这样得来的反事实条件句知识和模态知识,既非先验,亦非后验,应被归属为第三类的"椅验知识"(armchair thinking)。作者们认为威廉姆森的理论虽大体上可信,但仍至少有九个问题需要进一步探讨。     

Quantum Mechanics and Realism

Quantum mechanics is usually presented as a challenge to scientific realism, but I will argue that the details of quantum mechanics actually support realism. I will first present some basic quantum mechanical concepts and results, including the Einstein-Podolsky-Rosen (EPR) experiment and Bell's theorem, and do it in a way that everyone can understand. I will then use the physics to inform the philosophy, showing that quantum mechanics provides evidence to support epistemological realism.     

Quantum Nonlocality and the Challenge to Scientific Realism

In this essay I examine various aspects of the nearcentury-long debate concerning the conceptualfoundations of quantum mechanics and the problems ithas posed for physicists and philosophers fromEinstein to the present. Most crucial here is theissue of realism and the question whether quantumtheory is compatible with any kind of realist orcausal-explanatory account which goes beyond theempirical-predictive data. This was Einstein's chiefconcern in the famous series of exchanges with NielsBohr when he refused to accept the truth orcompleteness of a doctrine (orthodox QM) which ruledsuch questions to be strictly inadmissible. I discussthe later history of quantum-theoretical debate withparticular reference to the issue of nonlocality,i.e., the phenomenon of superluminal(faster-than-light) interaction betweenwidely-separated particles. Then I show how thestandard `Copenhagen' interpretation of QM hasinfluenced current anti-realist orontological-relativist approaches to philosophy ofscience. Indeed, there are clear signs that somephilosophers have retreated from a realist positionvery largely in response to just these problems. So itis important to ask exactly why – on what scientificor philosophical grounds – any preferred alternative(causal-realist) construal should have been ruled outas a matter of orthodox QM wisdom. Moreconstructively, my paper presents various arguments infavour of one such alternative, the `hidden-variables'theory developed since the early 1950s by David Bohmand consistently marginalised by proponents of theCopenhagen doctrine.     

论数学中的虚构主义

数学虚构主义是"数学实体不存在、数学中不存在真理、数学在世界的说明和科学事业中可有可无"的一种典型的当代数学反实在论的解释。通过对数学虚构主义的批判性分析,得出"数学虚构主义的反实在论规划整体上并不成功","数学在世界的说明和科学中是至关重要,而非可有可无"、"数学实体确实不存在"和"数学中存在真理,但其本质需进一步研究"的结论。