#logicism

[goddestroyer666]

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[omegaphilosophia]

The Philosophy of Arithmetic

The philosophy of arithmetic examines the foundational, conceptual, and metaphysical aspects of arithmetic, which is the branch of mathematics concerned with numbers and the basic operations on them, such as addition, subtraction, multiplication, and division. Philosophers of arithmetic explore questions related to the nature of numbers, the existence of mathematical objects, the truth of arithmetic propositions, and how arithmetic relates to human cognition and the physical world.

Key Concepts:

The Nature of Numbers:

Platonism: Platonists argue that numbers exist as abstract, timeless entities in a separate realm of reality. According to this view, when we perform arithmetic, we are discovering truths about this independent mathematical world.

Nominalism: Nominalists deny the existence of abstract entities like numbers, suggesting that arithmetic is a human invention, with numbers serving as names or labels for collections of objects.

Constructivism: Constructivists hold that numbers and arithmetic truths are constructed by the mind or through social and linguistic practices. They emphasize the role of mental or practical activities in the creation of arithmetic systems.

Arithmetic and Logic:

Logicism: Logicism is the view that arithmetic is reducible to pure logic. This was famously defended by philosophers like Gottlob Frege and Bertrand Russell, who attempted to show that all arithmetic truths could be derived from logical principles.

Formalism: In formalism, arithmetic is seen as a formal system, a game with symbols governed by rules. Formalists argue that the truth of arithmetic propositions is based on internal consistency rather than any external reference to numbers or reality.

Intuitionism: Intuitionists, such as L.E.J. Brouwer, argue that arithmetic is based on human intuition and the mental construction of numbers. They reject the notion that arithmetic truths exist independently of the human mind.

Arithmetic Truths:

A Priori Knowledge: Many philosophers, including Immanuel Kant, have argued that arithmetic truths are known a priori, meaning they are knowable through reason alone and do not depend on experience.

Empiricism: Some philosophers, such as John Stuart Mill, have argued that arithmetic is based on empirical observation and abstraction from the physical world. According to this view, arithmetic truths are generalized from our experience with counting physical objects.

Frege's Criticism of Empiricism: Frege rejected the empiricist view, arguing that arithmetic truths are universal and necessary, which cannot be derived from contingent sensory experiences.

The Foundations of Arithmetic:

Frege's Foundations: In his work "The Foundations of Arithmetic," Frege sought to provide a rigorous logical foundation for arithmetic, arguing that numbers are objective and that arithmetic truths are analytic, meaning they are true by definition and based on logical principles.

Russell's Paradox: Bertrand Russell's discovery of a paradox in Frege's system led to questions about the logical consistency of arithmetic and spurred the development of set theory as a new foundation for mathematics.

Arithmetic and Set Theory:

Set-Theoretic Foundations: Modern arithmetic is often grounded in set theory, where numbers are defined as sets. For example, the number 1 can be defined as the set containing the empty set, and the number 2 as the set containing the set of the empty set. This approach raises philosophical questions about whether numbers are truly reducible to sets and what this means for the nature of arithmetic.

Infinity in Arithmetic:

The Infinite: Arithmetic raises questions about the nature of infinity, particularly in the context of number theory. Is infinity a real concept, or is it merely a useful abstraction? The introduction of infinite numbers and the concept of limits in calculus have expanded these questions to new mathematical areas.

Peano Arithmetic: Peano's axioms formalize the arithmetic of natural numbers, raising questions about the nature of induction and the extent to which the system can account for all arithmetic truths, particularly regarding the treatment of infinite sets or sequences.

The Ontology of Arithmetic:

Realism vs. Anti-Realism: Realists believe that numbers and arithmetic truths exist independently of human thought, while anti-realists, such as fictionalists, argue that numbers are useful fictions that help us describe patterns but do not exist independently.

Mathematical Structuralism: Structuralists argue that numbers do not exist as independent objects but only as positions within a structure. For example, the number 2 has no meaning outside of its relation to other numbers (like 1 and 3) within the system of natural numbers.

Cognitive Foundations of Arithmetic:

Psychological Approaches: Some philosophers and cognitive scientists explore how humans develop arithmetic abilities, considering whether arithmetic is innate or learned and how it relates to our cognitive faculties for counting and abstraction.

Embodied Arithmetic: Some theories propose that arithmetic concepts are grounded in physical and bodily experiences, such as counting on fingers or moving objects, challenging the purely abstract view of arithmetic.

Arithmetic in Other Cultures:

Cultural Variability: Different cultures have developed distinct systems of arithmetic, which raises philosophical questions about the universality of arithmetic truths. Is arithmetic a universal language, or are there culturally specific ways of understanding and manipulating numbers?

Historical and Philosophical Insights:

Aristotle and Number as Quantity: Aristotle considered numbers as abstract quantities and explored their relationship to other categories of being. His ideas laid the groundwork for later philosophical reflections on the nature of number and arithmetic.

Leibniz and Binary Arithmetic: Leibniz's work on binary arithmetic (the foundation of modern computing) reflected his belief that arithmetic is deeply tied to logic and that numerical operations can represent fundamental truths about reality.

Kant's Synthetic A Priori: Immanuel Kant argued that arithmetic propositions, such as "7 + 5 = 12," are synthetic a priori, meaning that they are both informative about the world and knowable through reason alone. This idea contrasts with the empiricist view that arithmetic is derived from experience.

Frege and the Logicization of Arithmetic: Frege’s attempt to reduce arithmetic to logic in his Grundgesetze der Arithmetik (Basic Laws of Arithmetic) was a foundational project for 20th-century philosophy of mathematics. Although his project was undermined by Russell’s paradox, it set the stage for later developments in the philosophy of mathematics, including set theory and formal systems.

The philosophy of arithmetic engages with fundamental questions about the nature of numbers, the existence of arithmetic truths, and the relationship between arithmetic and logic. It explores different perspectives on how we understand and apply arithmetic, whether it is an invention of the human mind, a discovery of abstract realities, or a formal system of rules. Through the works of philosophers like Frege, Kant, and Leibniz, arithmetic has become a rich field of philosophical inquiry, raising profound questions about the foundations of mathematics, knowledge, and cognition.

[nicolae]

SetThings - Epistemologie, ontologie și logicism

https://www.setthings.com/ro/epistemologie-ontologie-si-logicism/

Epistemologie, ontologie și logicism

Dedekind și Frege: Epistemologiile lui Dedekind și ale lui Frege par mai puțin bine definite decât cele ale lui Russell, dar ambele par a accepta ca a priori „legile gândirii” obișnuite referitoare la afirmații propoziționale simple (de obicei ale credinței); … Read More

[nicolae]

SetThings - Logicism

https://www.setthings.com/ro/logicism/

Logicism

Logicismul este un program din filozofia matematicii, cuprinzând una sau mai multe dintre tezele că – pentru un anumit sens coerent al „logicii” – matematica este o extensie a logicii, o parte sau toată matematica poate fi redusă la logică … Read More

[siliconleaks-blog]

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24% #POSTMODERNISM

22% #LOGICISM

21% #LOGICAL_POSITIVISM

19% #OBJECTIVISM (#AYN_RAND)

9% #HEDONISM

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[isomorphismes]

A mind all logic is like a knife all blade. It makes the hand bleed that uses it.

Rabindranath Tagore, Stray Birds

[geopolicraticus]

The Two Philosophies of Mathematics

The early twentieth century (in other words, a hundred years ago) was a very exciting time in the philosophy of mathematics. The century before that had been punctuated by surprising mathematical discoveries and an unprecedented new level of rigor, and this fecundity gave rise to a similar fecundity in the philosophy of mathematics.

This is the period of thought that gave rise to the three classic schools of the philosophy of mathematics that appeared in countless books: logicism, formalism, and intuitionism. So why have I called this post “The Two Philosophies of Mathematics” instead of “The Three Philosophies of Mathematics”? Because each of these three philosophies of mathematics existed in two forms, and each form might well be isolated from the other and considered independently. I will refer to the dual aspect of each of these philosophies of mathematics as the formal aspect and the material aspect.

The formal aspect of logicism is the well-known reduction of mathematics to logic, so that every truth of mathematics can be formulated in purely logical terms, if one wants to take the trouble to write out the proof. The material aspect of logicism, however, was Frege’s conception of numbers as objects (a phrase that is also the title of a book by Crispin Wright). Thus the rigorous parsimony of the formal aspect of logicism was belied by a wonderful profusion of mathematical objects that characterized the material apsect of this school of thought.

The formal aspect of Hilbert’s formalism was the formulation of finitistic axiom systems for the whole of mathematics, which, like logicism, may also be understood as a form of reductionism, with infinitistic mathematics reduced to finite mathematics. The material aspect of formalism was the distinction Hilbert made between real and ideal mathematics, in order to be able to retain infinitistic mathematics, albeit in an altered form. Another material aspect of Hilbert’s thought is the idea that mathematics is a “game,” and thus not ultimately concerned with some stubborn reality, thus an early form of anti-realism and fictionalism.

The formal aspect of Brouwer’s intuitionism was the rejection of the law of the excluded middle, and, more generally, restrictions placed upon the methods considered acceptable in proof in order to  confine the mathematician to constructive methods. The material aspect of intuitionism was Brouwer’s mystical conception of mathematics as ultimately inexpressible in language, i.e., that language was ultimately a fall from grace, and the the deep sources of mathematical intuition derive from the experience of time.

In each case of these once powerful and influential philosophies of mathematics, then, there was a profound divergence between the formal and the material aspect -- less divergence in the case of Hilbert, more in the case of Russell. All of these different ideas, formal and material, have gone on to divergent careers in the history of thought. Logicism’s reduction of mathematics to logic became mainstream foundations of mathematics research, and merged with whatever remained of the formal aspect of Hilbert’s thought after Gödel proved the impossibility of realizing Hilbert’s program. Brouwer’s formal aspect blossomed into the embarras de richesses that is contemporary constructivism, which by a multiplicity of methods seeks to restrict the methods of mathematics. But the constructivists turned out to be the truly parsimonious school of thought in the long run, always seeking to double down on the profusion of mathematics, while the material aspect of logicism was transformed into contemporary Platonism, notwithstanding the fact that the origin of this is to be found in Russell, who was the greatest Ockhamist of his generation.

Can the distinction between formal philosophy of mathematics and material philosophy of mathematics be employed to understand the practice of philosophy of mathematics today, a hundred years after these classic philosophies played both sides of the fence, as it were? As noted above, foundations of mathematics research is almost entirely formal philosophy of mathematics, and only rarely do material ideas put in an appearance as thematic motives obscured by the focus on technique. This is not the relationship the obtained a hundred years ago, when formal and material ideas were not related as explicit and implicit aspects of the same conception. This this is largely descriptive of philosophy of mathematics today shows that there has been a certain convergence of formal and material ideas, and this convergence has resulted in a certain reduction in the diversity of philosophy mathematics. No one would compare the fecundity of a hundred years ago to the philosophy of mathematics scene today.

Perhaps this is the lesson to be learned: a period of great fecundity and diversity of thought, i.e., a period of Kuhnian revolutionary thought, can sustain a disconnect between formal and material ideas, while an extended period of “normal philosophy” in formal thought forces a kind of convergence and consensus.