Introduction

Henri Poincaré, a renowned French scientist from over a century ago, is still celebrated as one of the foremost intellectuals of his era. His expertise spanned across astronomy, mathematics, physics, and philosophy, earning him the title of the last universalist and polymath of his time. Some even attribute the foundations of relativity theory to him. As Emily Adlam notes, prior to Einstein, Poincaré had already formulated the correct Lorentz transformation equations, articulated the principle of relativity, derived the appropriate relativistic transformations for force and charge density, and established the rule for the relativistic composition of velocities (Emily Adlam, “Poincaré and Special Relativity”, 2011).

Poincaré was also deeply engaged in various social issues, notably supporting Alfred Dreyfus during the infamous Dreyfus affair by presenting statistical evidence related to Dreyfus's handwriting. This underscores Poincaré's significant influence and prominence as a scientist.

Imagine the challenge Bertrand Russell faced as he endeavored to lay the groundwork for mathematics while contending with such a prominent figure. This article will briefly explore some of Poincaré's remarks in Science and Method (1908) regarding Russell and the logicist movement.

The Early Criticisms of Poincaré Towards Russell and Other Philosophers of Mathematics

Poincaré's critique of logicism is encapsulated in his observations:

"Can mathematics be entirely distilled into logic without invoking principles unique to itself? There exists a whole school, fervent in their belief, dedicated to proving its feasibility. They have developed a specialized language, utilizing signs instead of words. This language is comprehensible only to a select few, causing the general populace to acquiesce to the authoritative declarations of its proponents." (Henri Poincaré, Science and Method, 143).

He adds:

"This approach contradicts sound psychological principles. It is unlikely that the human mind constructed mathematics in this manner. Is it logical or, more precisely, accurate? One can rightly question this." (Henri Poincaré, Science and Method, 144–145).

Poincaré criticized the emerging language of logicism for its ambiguity. While he acknowledged some aspects could be intriguing, his overall assessment was predominantly critical.

In Science and Method, he meticulously pointed out the contradictions faced by logicists, including Russell's paradox and Cantor’s paradoxes. Despite their attempts to resolve these contradictions, Poincaré cautioned that there was no guarantee that new contradictions would not emerge.

He remarked that the logicists:

"Have encountered contradictory conclusions, known as the Cantor antinomies. Despite this, they remain undeterred, modifying their principles to address previous contradictions, all while lacking assurance that new ones won't arise. It is time these exaggerations are addressed appropriately. Once we disprove one of their claims, it often resurfaces with minor alterations, reminiscent of the Lernaean Hydra with its ever-regenerating heads." (Henri Poincaré, Science and Method, 145).

Furthermore, Poincaré expressed concern over the logicists' assertion that their nascent models surpassed Kant's philosophy of mathematics:

"Mr. Russell and Signor Peano claim to have resolved the long-standing debate between Leibniz and Kant... They assert that mathematics is wholly reducible to logic, devoid of any role for intuition. Can we truly endorse such a sweeping judgment? I do not believe so." (Henri Poincaré, Science and Method, 146).

He succinctly summarized his views:

"Mr. Russell and Mr. Hilbert have both exerted considerable effort and produced works rich with original and often profound insights... However, to claim they have definitively settled the dispute between Kant and Leibniz and dismantled Kant’s mathematical theory is clearly untrue." (Henri Poincaré, Science and Method, 176).

Poincaré could not reconcile the idea of mathematics being derived solely from logical concepts. While it is uncertain if Russell and Couturat addressed Kant’s philosophy, Frege did present compelling arguments against Kant, highlighting the ongoing debate.

Poincaré further noted:

"Even with complete success, would Kantians be silenced? It is likely not, as one cannot reduce mathematical thought to a mere empty form without distortion. Even if it were established that all theorems could be derived through purely analytical processes from a finite set of axioms—axioms that are essentially conventions—a philosopher would still have the right to question the origins of those conventions and why they are deemed preferable over alternatives." (Henri Poincaré, Science and Method, 148).

He concluded:

"Philosophers who refuse to accept that logic is the entirety of the matter would undertake a noble task. My goal is to ascertain whether, once the principles of logic are accepted, we can demonstrate all mathematical truths without invoking intuition again." (Henri Poincaré, Science and Method, 149).

In summary, Russell aimed to:

"Construct the entirety of mathematics without introducing any new elements." (Henri Poincaré, Science and Method, 164).

Is this truly feasible? This question continues to fuel debate among philosophers of mathematics today.

My Interpretation of His Comments

The logicist movement sought to formalize mathematics and logic as thoroughly as possible, a pursuit that persists. However, it appears that significant advancements in the philosophy of mathematics remain elusive from that era to the present.

Researchers continue to explore various avenues, but should they be labeled "Hydras" due to their missteps along the way? Referring to them as ‘Hydras’ is a poetic yet harsh critique of their endeavors.

A fair examination reveals several key points:

• Cantor crafted a highly expressive logical framework capable of articulating any reasoning, whether in natural or formal languages. A set, for example, can represent a group, class, concept, world, or universe, depending on interpretation.
• Frege made remarkable strides in defining natural numbers using Cantor’s language. His definition remains so robust that the community is still grappling with its shortcomings. Claims of obvious errors in his work often mislead.
• Paradoxes in Cantor’s set theory and Gödel’s conclusions demand ongoing scrutiny. Graham Priest's dialetheism offers insights into these paradoxes, while Wittgenstein expressed dissatisfaction with Gödel’s reasoning—an unresolved debate.
• The interplay between logic, mathematics, and philosophy continues to spark contention, with no consensus on their boundaries. This ambiguity may be intrinsic to the nature of these disciplines, or it may lead to more definitive insights in areas like dialetheism, the Gödel-Wittgenstein debate, and foundational set theory.

Final Statements

Poincaré's early critiques of Russell, along with his views on logicism and formalism, were notably harsh. However, in his Last Essays (1913), he shifted focus to the limitations of logic and the freedom of mathematics beyond these constraints.

His insights on Russell and other mathematical philosophers were indeed thought-provoking and significant, yet it is curious that his philosophy of mathematics is seldom studied within the field of logic.

Is it due to his intimidating presence? Or perhaps his writing in French rather than English? His early 20th-century passing may have left him without a chance to finalize his arguments. Could his critiques have prompted some form of resistance? Or is it simply that many scholars are uninterested in exploring beyond formal structures? Numerous possibilities exist.

References

• Henri Poincaré. Science and Method. Ed. Cosimo Classics, 2007.
• Henri Poincaré. Mathematics and Science: Last Essays. Kessinger Publishing, 2010.
• Heinzmann, Gerhard and David Stump. “Henri Poincaré”. The Stanford Encyclopedia of Philosophy (Summer 2024 Edition), Edward N. Zalta & Uri Nodelman (eds.).
• Irvine, Andrew David. “Bertrand Russell”. The Stanford Encyclopedia of Philosophy (Spring 2022 Edition), Edward N. Zalta (ed.).