Algebra As Thought Experiment

1998 ◽  
pp. 106-111
Author(s):  
Jagdish Hattiangadi
Keyword(s):  
Experimental Investigation ◽  
Thought Experiment ◽  
A Priori ◽  
Modern Science ◽  
Algebraic Method ◽  
Mathematical Physics ◽  
Physical World ◽  
Empirical Science ◽  
The World ◽  
Synthetic A Priori

This paper addresses the problem of understanding what mathematics contributes to the exceptional success of modern mathematical physics. I urge that we give up the Kantian construal of the division between mathematics (synthetic a priori) and physics (experimental), and that we ask instead how algebra helps synthetic a posteriori mathematics improve our ability to study the world. The theses suggested are: 1) Mathematical theories are about the empirical world, and are true or false just like other theories of empirical science. 2) The air of artificiality in mathematics lies exclusively in the use of algebraic method. 3) This method is constructive much like all fiction is, but this construction is for the purpose of experimental investigation of the physical world to the extent that anything in the world has objects like those in the fictional world of a particular algebra. 4) This is why algebraic techniques are successful even when the assumptions of the system are false: they may still be applicable to some things considered from some perspective. 5) The success of mathematical physics is also due to Descartes' discovery of a remarkable truth: we live in space and time which can be described as a whole. 6) Therefore, what distinguishes modern science from earlier and later philosophy is not a general method of science, but the fact that it happened to find a truth, and a particular way of studying reality which bore fruit.

Reichenbach, Hans (1891–1953)

2018 ◽  
Author(s):  
Wesley C. Salmon
Keyword(s):  
Twentieth Century ◽  
A Priori ◽  
Vienna Circle ◽  
Philosophy Of Physics ◽  
Theory Of Relativity ◽  
Sense Experience ◽  
Space And Time ◽  
The World ◽  
Synthetic A Priori ◽  
And Mathematics

Philosophy of science flourished in the twentieth century, partly as a result of extraordinary progress in the sciences themselves, but mainly because of the efforts of philosophers who were scientifically knowledgeable and who remained abreast of new scientific achievements. Hans Reichenbach was a pioneer in this philosophical development; he studied physics and mathematics in several of the great German scientific centres and later spent a number of years as a colleague of Einstein in Berlin. Early in his career he followed Kant, but later reacted against his philosophy, arguing that it was inconsistent with twentieth-century physics. Reichenbach was not only a philosopher of science, but also a scientific philosopher. He insisted that philosophy should adhere to the same standards of precision and rigour as the natural sciences. He unconditionally rejected speculative metaphysics and theology because their claims could not be substantiated either a priori, on the basis of logic and mathematics, or a posteriori, on the basis of sense-experience. In this respect he agreed with the logical positivists of the Vienna Circle, but because of other profound disagreements he was never actually a positivist. He was, instead, the leading member of the group of logical empiricists centred in Berlin. Although his writings span many subjects Reichenbach is best known for his work in two main areas: induction and probability, and the philosophy of space and time. In the former he developed a theory of probability and induction that contained his answer to Hume’s problem of the justification of induction. Because of his view that all our knowledge of the world is probabilistic, this work had fundamental epistemological significance. In philosophy of physics he offered epoch-making contributions to the foundations of the theory of relativity, undermining space and time as Kantian synthetic a priori categories.

Thought Experiments

Philosophy ◽  
2013 ◽  
Author(s):  
James Robert Brown ◽  
Michael T. Stuart
Keyword(s):  
Thought Experiment ◽  
A Priori ◽  
Thought Experiments ◽  
Widespread Agreement ◽  
The World ◽  
Set Up ◽  
The Mind ◽  
Insight Into

Thought experiments are performed in the imagination. We set up some situation, we observe what happens, then we try to draw appropriate conclusions. In this way, thought experiments resemble real experiments, except that they are experiments in the mind. The terms “thought experiment,” “imaginary experiment,” and “Gedankenexperiment” are used interchangeably. There is no consensus on a definition, but there is widespread agreement on which are standard examples. It is also widely agreed that they play a central role in a number of fields, especially physics and philosophy. There are several important questions about thought experiments that naturally arise, including what kinds of thought experiments there are, what roles they play, and how, if at all, they work. This last question has been the focus of much of the literature: How can we learn something new about the world just by thinking? Answers range from “We don’t really learn anything new” to “We have some sort of a priori insight into how nature works.” In between there are a great variety of rival alternative accounts. There is still no consensus; debate is wide open on almost every question pertaining to thought experiments.

Are Thought Experiments Just What You Thought?

1996 ◽  
Vol 26 (3) ◽  
pp. 333-366 ◽  
Author(s):  
John D. Norton
Keyword(s):  
Public Relations ◽  
Scientific Literature ◽  
Thought Experiment ◽  
Thought Experiments ◽  
Physical World ◽  
Science Writing ◽  
Experimental Science ◽  
Original Intent ◽  
The World

Whatever the original intent, the introduction of the term ‘thought experiment’ has proved to be one of the great public relations coups of science writing. For generations of readers of scientific literature, the term has planted the seed of hope that the fragment of text they have just read is more than mundane. Because it was a thought experiment, does it not tap into that infallible font of all wisdom in empiricist science, the experiment? And because it was conducted in thought, does it not miraculously escape the need for the elaborate laboratories and bloated budgets of experimental science?These questions in effect pose the epistemological problem of thought experiments in the sciences:Thought experiments are supposed to give us information about our physical world. From where can this information come?One enticing response to the problem is to imagine that thought experiments draw from some special source of knowledge of the world that transcends our ordinary epistemic resources.

Color Primitivism and Neo-pragmatism

2017 ◽  
Author(s):  
Joshua Gert
Keyword(s):  
Natural Kinds ◽  
A Priori ◽  
Empirical Science ◽  
The World ◽  
Color Primitivism ◽  
Descriptive Language

This chapter responds to criticisms raised by Jonathan Cohen, on behalf of reductionists, to the Benacerraf-style argument for color primitivism offered in Chapter One. The response stresses the fact that the argument for primitivism is perfectly consistent with the idea that some ostensively taught terms—terms for natural kinds, for example—refer to properties that have hidden essences that are the business of empirical science to determine. In this way, the Benacerraf-style argument is perfectly consistent with the idea that water is identical to H2O. The chapter also presents in much more detail the neo-pragmatism on which the book relies throughout. Rather than making the a priori assumption that descriptive language must function by making use of words that “latch on” via a substantive relation of reference to objects and properties out there in the world, the neo-pragmatist takes a more empirical view of language that reflects a deeper naturalism.

Kreines on the Problem of Metaphysics in Kant and Hegel

2016 ◽  
Vol 39 (1) ◽  
pp. 106-120
Author(s):  
Robert Stern
Keyword(s):  
A Priori ◽  
Transcendental Idealism ◽  
A Priori Knowledge ◽  
The World ◽  
Synthetic A Priori ◽  
Priori Knowledge

AbstractThis article offers a discussion of James Kreines’s book Reason in the World: Hegel’s Metaphysics and Its Philosophical Appeal. While broadly sympathetic to Kreines’s ‘concept thesis’ as a conceptual realist account of Hegel, the article contrasts two Kantian arguments for transcendental idealism to which Hegel’s position may be seen as a response—the argument from synthetic a priori knowledge and the argument from the dialectic of reason—and explores the implications of Kreines’s commitment to the latter over the former.

Descartes on God's relation to time

2008 ◽  
Vol 44 (4) ◽  
pp. 413-431 ◽  
Author(s):  
GEOFFREY GORHAM
Keyword(s):  
First Principles ◽  
A Priori ◽  
Mathematical Physics ◽  
Physical World ◽  
Recent Commentary ◽  
Strong Current ◽  
Before And After ◽  
Important Barrier ◽  
Seamless Connection

AbstractGod and time play crucial, intricately related roles in Descartes' project of grounding mathematical physics on metaphysical first principles. This naturally raises the perennial theological question of God's precise relation to time. I argue, against the strong current of recent commentary, that Descartes' God is fully temporal. This means that God's duration is successive, with parts ordered ‘before and after’, rather than permanent or ‘all at once’. My argument will underscore the seamless connection between Descartes' theology and his physics, and the degree to which he was prepared to depart from orthodoxy in the former in order to secure an a priori foundation for the latter. As Newton would later do, Descartes freed time from its traditional dependence on bodily motion and so removed an important barrier to making God temporal. Acting in time, God makes the physical world intelligible in a way He could not were He timeless.

scholarly journals Machines, Logic and Quantum Physics

2000 ◽  
Vol 6 (3) ◽  
pp. 265-283 ◽  
Author(s):  
David Deutsch ◽  
Artur Ekert ◽  
Rossella Lupacchini
Keyword(s):  
Quantum Physics ◽  
A Priori ◽  
Physical Reality ◽  
Physical World ◽  
Empirical Science ◽  
Physical Sciences ◽  
Mathematical Structures ◽  
Eugene Wigner ◽  
The Universe ◽  
Modern Status

§1. Mathematics and the physical world. Genuine scientific knowledge cannot be certain, nor can it be justified a priori. Instead, it must be conjectured, and then tested by experiment, and this requires it to be expressed in a language appropriate for making precise, empirically testable predictions. That language is mathematics.This in turn constitutes a statement about what the physical world must be like if science, thus conceived, is to be possible. As Galileo put it, “the universe is written in the language of mathematics”. Galileo's introduction of mathematically formulated, testable theories into physics marked the transition from the Aristotelian conception of physics, resting on supposedly necessary a priori principles, to its modern status as a theoretical, conjectural and empirical science. Instead of seeking an infallible universal mathematical design, Galilean science usesmathematics to express quantitative descriptions of an objective physical reality. Thus mathematics became the language in which we express our knowledge of the physical world — a language that is not only extraordinarily powerful and precise, but also effective in practice. Eugene Wigner referred to “the unreasonable effectiveness of mathematics in the physical sciences”. But is this effectiveness really unreasonable or miraculous?Numbers, sets, groups and algebras have an autonomous reality quite independent of what the laws of physics decree, and the properties of these mathematical structures can be just as objective as Plato believed they were (and as Roger Penrose now advocates).

Is There a Scientific Method?

2021 ◽  
pp. 21-35
Author(s):  
Steven L. Goldman
Keyword(s):  
Subjective Experience ◽  
Deductive Reasoning ◽  
A Priori ◽  
Inductive Reasoning ◽  
Modern Science ◽  
Francis Bacon ◽  
Human Reasoning ◽  
The World ◽  
Innate Ideas ◽  
External Causes

Modern science relies largely on method or, rather, on the claim that by employing a systematic, impersonal method, human reasoning can transcend the mind’s subjective experience of reality and discover the true, external causes of experience. In the early stages of modern science’s emergence out of medieval and Renaissance nature philosophy, Francis Bacon argued that this method was to be based on induction and experiment, without a priori mental input and with a minimum of mathematics. Rene Descartes argued that the required method was to be based on deduction, mathematics, and a priori and innate ideas, with a minimum of experiment. For Descartes, experiment served primarily as a check on deductive reasoning; for Bacon, experiment was a source of knowledge and constrained our inductive reasoning about empirical facts. Despite their differing styles, Descartes and Bacon together concretized the idea that a systematic method of reasoning could give us knowledge of the world.

Transcendental Arguments

Philosophy ◽  
2017 ◽  
Author(s):  
Gabriele Gava
Keyword(s):  
A Priori ◽  
The Other ◽  
Necessary Condition ◽  
Analytical Relationship ◽  
Critique Of Pure Reason ◽  
Transcendental Arguments ◽  
Cambridge University ◽  
The World ◽  
Synthetic A Priori ◽  
Historical Antecedents

As generally understood, transcendental arguments are deductive arguments that aim to establish a certain claim A by arguing that A is a necessary condition for another claim B. Customarily, they are used to refute various forms of skepticism. Accordingly, B is usually a claim that is noncontroversial and would plausibly be accepted by a skeptic: for example, the claim that we have self-consciousness, or that we have representations of objects. Alternatively, B could also be a claim that a skeptic must assume to coherently formulate her doubt. Transcendental arguments then proceed from this noncontroversial claim to a more substantial claim that states that A is a necessary condition for the possibility of B. The skeptic who doubts that A applies but accepts B is thus refuted because, if B applies, it logically follows that A must apply as well. Debates about transcendental arguments have touched on a multiplicity of issues. One first question concerns the nature of the claims they make. In this respect, there have been different ways to account for the necessity that is attributed to the claims that are identified as conditions of other claims. While it is excluded that this necessity can be physical or causal, it is not clear what kind of necessity it is. Some have claimed that this necessity expresses analytical relationship between concepts, whereas others have understood this necessity to be of a metaphysical nature and to involve some sort of synthetic a priori judgment. Another problem concerns what kind of results transcendental arguments can achieve. Some have claimed that transcendental arguments can achieve ambitious conclusions that tell how the world must be. Others have presented a more modest interpretation of transcendental arguments, claiming that they can establish only how we must believe the world to be. A further issue regards the historical antecedents of contemporary transcendental arguments. While Kant is normally considered to be the originator of transcendental arguments, it has been questioned that central arguments of his Critique of Pure Reason (Cambridge: Cambridge University Press, 1998) have a structure fundamentally similar to present-day transcendental arguments. On the other hand, arguments with a form comparable to transcendental arguments have been attributed to other philosophers and traditions.

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