Truth, in the eyes of Foucault, is a mirror of power. In Power/Knowledge, Foucault mentions: “Each society has its regime of truth, its 'general politics' of truth: that is, the types of discourse which it accepts and makes function as true; the mechanisms and instances which enable one to distinguish true and false statements, the means by which each is sanctioned; the techniques and procedures accorded value in the acquisition of truth; the status of those who are charged with saying what counts as true.” It is through the discourse of truth that power determines itself, as a constantly changing mechanism that produces the division between truth and untruth. The relationship between truth and power forms the very fabric of our society as an intricate network of discourse. This “regime of truth” seeks to diminish and nullify other forms of statements, separating the rational from the irrational. This web of power flows ceaselessly, shaping and being shaped until they become what supports the system; they are invisible, permeating every aspect of our society. Foucault notes, “The important thing here, I believe, is that truth isn't outside power, or lacking in power… Truth is a thing of this world: it is produced only by virtue of multiple forms of constraint.”
On the other hand, the truth we refer to in the context of mathematical proofs appears to bear a different meaning. Questions such as whether there are infinitely many primes seem to be exempt from the constraints of societal power structures. They seem to transcend beyond the bounds of the power structure and societal constraints. Throughout history, there has been a plurality of ways in which people define truth, or propose a definition of truth. For instance, early twentieth-century philosophers such as Bertrand Russell and Gottlob Frege, had proposed different approaches such as the correspondence principle or the coherence principle among others. However, it seems that Foucault is simply not interested in this discussion. Naturally, a question arises about how to reconcile Foucault's claim that "truth isn't outside of power" with mathematical truth.
In the first half of the essay, I will begin by examining Foucault's notion of truth in its relationship with power and knowledge. The second half will shift the focus to Foucault's relation to mathematical truth, exploring his connection with the aesthetics of living and Jean Cavaillès' idea that mathematics is a becoming. Finally, I will highlight the potential issue of interpreting Foucault without considering his intentions and suggest that we should approach his work in a constructive manner.
Among many of his other works, in The Discourse on Language, Foucault identifies the will to truth as a third system of external exclusion, along with prohibited words and madness. Together, these three systems divulge the forces that separate what is included in the system that is considered rational from what is not. In summary, Foucault suggests in this essay that this division of true and false is at the core of the will to truth, or will to knowledge, which has survived for two thousand years. Tracing back to the ancient Greeks in the sixth century, the fabric of this mechanism is reinforced by the performative ritual that both predicts and determines the unforeseen future, which became the so-called fate itself that governs the occurrence of events. A century later, truth evolved into what is being said, without the performative aspect of ritual and determining events as reinforcement. It became static and enunciated itself, but it was no longer desirable since it became disjointed with the exercise of power. At the same time, Foucault points out the dynamic quality of the will to truth– “it has never ceased shifting.” He believes that the will to truth carries with it a history of its own, separate from the history of truth itself. This history includes “the history of a range of subjects to be learned, the history of the functions of the knowing subject, the history of material, technical, and instrumental investment in knowledge.” The will to truth, he asserts, relies on institutional support and wields power over other forms of discourse. It is reinforced and accompanied by various practices, such as pedagogy, the book system, publishing, libraries, and learned societies. As these practices have evolved over time, Foucault suggests that the words of the law on longer hold authority in society, except insofar as they are derived from true discourse. Compared to the other two forms of exclusion, the will to truth appears as a more stable form, exerting more power and control, as it gradually eliminates and nullifies other forces to reaffirm its own stable foundation. He concludes, “Thus, only one truth appears before our eyes: wealth, fertility, and sweet strength in all its insidious universality. In contrast, we are unaware of the prodigious machinery of the will to truth, with its vocation of exclusion.” Therefore, by bringing forward the discussion on the will to truth, Foucault unveils the ever-changing machinery that governs the truth and the will of truth, which has been overlooked throughout history.
In Power/Knowledge, Foucault asserts “truth isn’t outside of power,” and ”truth is the thing of this word.” He suggests, 'the economic policy of Truth' is centered on the scientific discourses and the institutions which produces it, it circulates through apparatuses such as education and information, it is regulated and controlled by political and economic apparatuses such as university, and media. He sees a circular relationship between truth and the system of power that produces it– one produces, sustains, and reinforces the other. This circular relationship is considered “the regime of truth,” it is a system for “production, regulation, distribution, circulation and operation of statements,” which, as a system, enables one to distinguish true statements from false.
Although Foucault acknowledges that his thinking pre-assumes that power ‘is always already there’, one is never ‘outside’ it, and there is no ‘margin’ for those who break the system to gambol in, he does not agree to the thinking such that Individual is trapped within power. He also challenges the idea that power merely serves the function of prohibitions and exclusion, or that power is primarily in service of an economic interest. He rather sees power taking multiple forms, as a precondition of one to exist in society, in which no primal liberty exists outside of this mesh. In other words, he sees power as an integrated, interwoven, unitary body co-existing with the social body, domination emerging out of this web, other than seeing it as a binary domination between two parties. 
So far, we have discussed Foucault's perspective on the relationship between truth and power. We can discern a pattern that he views all forms of knowledge through the lens of power. However, this way of filtering knowledge can itself function as a system of power and domination. In the discussion on mathematics and physics, he considers them as either products of power or as resembling power structures. For Foucault, mathematics and physics are embodiments of a sense of hierarchical separation and classical rationalist ideology that separates them from other disciplines. Foucault suggests, “In the field we are concerned with here, it followed that they wanted to take up the 'noblest', most academic problems in the history of the sciences: mathematics and physics, in short the themes valorised by Duhem, Husserl and Koyre. Medicine and psychiatry didn't seem to them to be very noble or serious matters, nor to stand on the same level as the great forms of classical rationalism.” In another example, Foucault views different subsets of mathematics as symbolic of two political structures: one democratic and the other totalitarian. Foucault notes: “It is worth recalling at this point, if only symbolically, the old Greek adage, that arithmetic should be taught in democracies, for it teaches relations of equality, but that geometry alone should be served for oligarchies, as it demonstrates the proportions within inequality.”  
Similarly, the will to truth for Foucault is filtered into a binary division that exists within societal power structures. Truth became a product of exclusion and power plays that take place within these structures. The pursuit of "truth" and "the will to truth" has become synonymous with a will to divide and exclude– an active process that has an inherent tendency of division and exclusion in service of authority and power. In The Discourse on Language, Foucault mentioned, “Certainly, as a proposition, the division between true and false is neither arbitrary, not modifiable, nor institutional, nor violent. Putting the question in different terms, however– asking what has been, what still is, throughout our discourse, this will to truth which has survived throughout so many centuries of history; or if we ask what is, in its very general form, the kind of division governing out will to knowledge– then we may well discern something like a system of exclusion (historical, modifiable, institutionally constraining) in the process of development.” Here, there appears to be an acknowledgment of the propositional separation between true and false that exists independently of institutional power. However, the focus primarily centers on the exclusionary forces that are contingent on history and society. In other words, while the existence of a propositional separation between true and false is recognized, it is simultaneously disregarded. Consequently, only the dynamic and structural aspects of power remain visible, while the question of whether there is a propositional separation between true and false seems to be both acknowledged and ignored.
In another example, Foucault constructs a tautology. He uses the definition of truth as a purely social and historical contingency to emphasis that all truths are contingent on social and historical factors. He argues that the notion of truth is always intertwined with power and is never outside of its influence, yet defines truth as such. He suggests “There is a battle 'for truth', or at least 'around truth'- it being understood once again that by truth I do not mean 'the ensemble of truths which are to be discovered and accepted', but rather 'the ensemble of rules according to which the true and the false are separated and specific effects of power attached to the true', it being understood also that it's not a matter of a battle 'on behalf' of the truth, but of a battle about the status of truth and the economic and political role it plays.” He defined what he meant by truth as "the ensemble of rules according to which the true and the false are separated and specific effects of power attached to the true.” However, his definition of truth serves precisely the purpose of revealing that there is no other form of truth that exists outside of power. This underlying tautology exercises power by creating a self-enclosed structure that enhances its own power. By limiting the discussion of truth to social and historical conditions, it can potentially exclude the consideration of truth that exists outside of societal constraints. As a result, other forms of discourse on truth (i.e on mathematics, morality, beauty and etc...) may be repelled and dismissed as irrational.
In our discussion on Foucault's views on mathematical truth, there seems to be a void in his views regarding mathematical and logical truth. He often shifts the focus of the discussion towards power or uses mathematical truth as an example to illustrate power dynamics. He creates tautologies that emphasize his intentions and definition of truth, which can potentially be seen as an authoritative view. In this case, how should one situate this lack of answer and what is the condition behind his attitude on truth?
First of all, we can look at Foucault's philosophy towards life -- the aesthetic of living. Instead of looking at the truth itself, Foucault spent a lot of attention at the speaking of truth – the parrhesia. In ancient Greek, courage is a proof of one who speaks the truth, indicating by what they say what is different from the majority. Foucault analyzes truth-telling activity as an important role in ancient Greek society, to which this activity is highly tied to the aesthetic of life. They are linked together through Socratic dialogue, and it became a way of living. In other words, Foucault’s keen interest in the speaking of truth constitutes the meaning of truth as something that is indistinguishable from the way of living as an individual in a society. 
To provide a little background, since Socrates, the philosophical discussion of the division between the soul and the living has been united into one. However, after Socrates' death, his philosophy became divided into two ways: Socrates- Platonism versus Socrates- Antisthenesism. Plato established his academy to discuss pure criticism and theories, while Antisthenes established Cynicism, which emphasizes the ways of living and practicing life. Since then, the Western tradition of metaphysics, analytic theory, methodology, and criticism has been established by Plato's investigation of Socrates' philosophy about the soul. In this tradition, the purpose is to answer questions about the self and the soul, in the sense that this soul exists in a world of ultimate reality beyond daily existence, beyond the physical world. Which means one can not discuss this line of thinking without positing a metaphysical realm of truth, the other world to situate the soul. On the other hand, what Antisthenes established is a philosophy about a way of living. Through this way of living, one directly accesses truth and authenticity– parrhesia and removes the unnecessary parts. 
Foucault believes the philosophy of the soul, as represented by Plato, has dominated the Western philosophical tradition, while the philosophy of living, represented by Antisthenes, has been ignored and forgotten. For Foucault, the philosophy of living is to treat life as an aesthetic subject, which relates to his 'practices of the self.' He believes that the aesthetic of life has been suppressed in Western discourse through the emphasis on objecthood in art, along with the ontology of the soul that prioritizes criticism, theory, and metaphysical truth. Foucault notes: “What strikes me is the fact that in our society, art has become something which is related only to objects and not to individuals, or to life. That art is something which is specialized or which is done by experts who are artists. But couldn't everyone's life become a work of art? Why should the lamp or the house be an art object, but not our life?” 
Therefore, we can discern that Foucault's intention in defining truth as contingent upon social, historical, and political factors is to echo his sympathy for Antisthenesism, which emphasizes the importance of living over ideal rationality. By rejecting the a priori notion of truth, Foucault symbolically demolishes the dominant way of thinking in the Western discourse, represented by Plato as the source that prioritizes discourse on the soul, rationality, dialectics, and distinguishing truth from non-truth. His aim is to subvert the uneven balance between knowledge that is marginalized by Western paradigms and the dominant ones. Unlike Kant, who attempts to establish a foundation of self-autonomy through proposing an a priori notion of truth grounded in the acknowledgment of each moral agent as a rational being and emphasizing universalized maxims, Foucault intends to establish a foundation for self-autonomy through each member of society's action of tracing back into history to acknowledge the limitations imposed upon them and the possibility of transcending such limits. In other words, for Foucault, an emphasis on the self-evidency of logical or mathematical truth is a symbol of the domination of Platonism, which explains why he does not extensively discuss them.
The shift from a focus on knowledge to a focus on life enabled Foucault to draw attention to the role of the individual as an agent in society. The purpose of discussing truth is both to criticize how the uses of truth are composed of and weighted by historical value and power mechanism, but also to establish a normative value of truth that is situated within society that emphasizes self-autonomy through examining history. However, the question is still remains unsolved– how do we reconcile the two notions of truth?
To construct a possible response from Foucault, his influences from Jean Cavaillès' thinking provide a potential perspective on his relationship with the foundations of mathematics. We can observe many resemblances between Cavaillès and Foucault's thinking, particularly regarding the emphasis on concepts rather than universal consciousness and the finding of historical dimension in Mathematics. As Foucault points out, Cavaillès, along with Brunschvicg, Bachelard, Koyre, and Canguilhem, introduces an approach to studying experience that isn't based on a foundation in the subject. Cavaillès’ projects were, in a way, also aimed at establishing the status of self-autonomy and rationality (similar to Kant). However, unlike Kant's approach, where the a-priori notion of truth in mathematics is universal, pre-assumed and determined, Cavaillès detaches rationality from man's consciousness like Husserl and connects mathematics to historical contingency. Cavaillès suggests, “Mathematics is a becoming. All we can do is try to understand its history; that is to say, to situate mathematics in relation to other intellectual activities, to discover certain characteristics of this becoming.” In other words, mathematics for him is not determined in any discursive framework or set of evidence external to itself. Even the most elemental elements in mathematics are the outcome of predecessor acts which do not have a simple origin. Cavaillès acknowledges the Hiberlian sign as a constructive object which granted the status of mathematics activity as detach of empirical reality. Meanwhile in considering Gödel's incompetent theorem, he does not embrace the thinking that one can find a foundation, or a consistent set of axioms that can include all possible mathematical expressions. In other words, there is no a-priori notion of truth determine which axiom is the true axiom. In this case, he believes the foundation for mathematics should always be able to alter and evolve, as a constant ‘becoming’ through historical contingency. This approach thus for Cavaillès, successfully preserved the contingency and creative elements in mathematics while considering it fundamental. Therefore we can consider Cavaillès’ thinking allows historical contingency to coexist with the transcendental and the timeless. In light of which, a notion of truth which contingency on social and historical context can be seen as a symbol for dismantling the tradition of philosophy rooted in subjective consciousness and the projection of truth as universal, generic, reductive, and incessant, while still preserving the necessary conditions underlying it. 
However, it's worth pointing out that the assertion that even mathematical truth is a product of this world or contingent on history could be misconstrued as suggesting that what makes mathematical knowledge true and how we understand mathematical knowledge rely on history. In reality, what is shaped by history is mathematics as a subject—a field that determines which axioms to use and which topics to study. However, once we establish an axiom, the internal consistency and coherence within that axiom are entirely independent of historical factors. Therefore, we should not interpret Cavaillès' claim that mathematics is historical in a literal sense and consider it as a foundational aspect of Foucault's notion of truth.
With that being said, we, as readers of Foucault, must trace back to the foundations behind Foucault. His notion of truth has been well-popularized in many aspects of society, circulating through constant repetition and variation. Without considering his intention, there is a potential danger to perceive his philosophy as pure nihilism– a negative instance of complete mistrust in our systems of knowledge and understanding. In fact, the cynicism that Foucault was interested in has been altered and infiltrated the very fabric of our culture. This sense of cynicism, through different generations of iterations, has already detached from what was initially meant, transforming into a pervasive distrust and rejection of ethical and social values. This pessimistic view of humanity's ability to make ethical choices stands in direct opposition to Foucault's intended purpose. 
Therefore, one needs to be careful not to see Foucault as an agent of authority. In The Discourse on Language, Foucault identifies authorship as one of the three types of internal exclusion, along with commentary and discipline. According to Foucault, an author is "the unifying principle in a particular group of writings or statements, lying at the origins of their significance, as the seat of their coherence." It is a tendency to attribute a body of work to an author, a persona that we invent based on what we know about their life, beliefs, and opinions. By doing so, we may attribute certain ideas to the author without looking at the context and condition behind them. Foucault, one of the most influential thinkers of the 20th century, can be himself subjected to the attribute of an author with authority, which means one may consider the danger of granting him undue authority as an agent of power. 
This is not to suggest that we should adopt a completely opposing view to Foucault on truth, nor should we underestimate the value of his thinking in any way. As Foucault himself acknowledge, one must constantly question the underlying assumptions and blind spots of one’s thinking. That allows us to employ the examination of mathematical truth as a lens to explore the conditions and limitations underpinning Foucault's own perspective. In my view, when examining Foucault's work, we should focus on the constructive aspect he proposes— the notion of practicing life as an ongoing examination to establish self-autonomy, in contrast to an interpretation lead to the modern notion of cynicism, which entails a sense of pessimistic distrust of the system of knowledge. Thus, the absence of discussion about mathematical truth serves as a gateway to glimpse what Foucault truly stands for in terms of metaphysics, epistemology, aesthetics, ethics, and the roles of humanity and rationality. Not seeing Foucault as an author in turn reveals both the essence and limitation beyond mere appearance. 

Foucault, M., & Gordon, C. (2015). Power/Knowledge: Selected interviews and other writings 1972-1977. Vintage Books. P131
Foucault, M., (1987). The Discourse on Language. essay, Dorset Press. p215-237 
What is Enlightenment? in Rabinow (P.), éd., The Foucault Reader, New York, Pantheon Books, 1984, pp. 32-50 
The Meaning and Evolution of the Word “Parrhesia”: Discourse & Truth, Problematization of Parrhesia - Six lectures given by Michel Foucault at the University of California at Berkeley, Oct-Nov. 1983 
汪民安. (2020). 福柯最后的哲学思想. 中国人民大学学报, 142–151.
Foucault, M., & Rabinow, P. (1984). The foucault reader, On the Genealogy of Ethics: An Overview of Work in Progress. Pantheon Books. p341
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