What makes this book truly shine is Kucharski’s gift for making the abstract feel tangible. He effortlessly bridges disciplines—from ancient philosophy to modern cryptography, from courtroom trials to scientific experiments—highlighting how humans have always wrestled with questions of proof, truth, and trust. The result is a narrative that is not only educational but deeply engaging.
Kucharski doesn’t just present facts; he provokes reflection. At a time when misinformation abounds and public trust in expertise is under strain, Proof arrives as both a timely and timeless work. It reminds us that certainty is often less about having all the answers and more about asking the right questions—and understanding the tools we use to find our way toward truth.
Whether you're a science enthusiast, a philosopher at heart, or simply someone who enjoys a good intellectual detective story, Proof offers insight, clarity, and just the right amount of challenge. Adam Kucharski has written a book that is not only readable but re-readable—a true testament to the power of thoughtful inquiry.

Every so often I need to dip into philosophy, mix in some math, and sprinkle on some science. Give me that interdisciplinary trip, please and thank you! Proof: The Uncertain Science of Certainty, by Adam Kucharski, seemed like just what Dr. Kara ordered for herself. Indeed, it’s illuminating enough, though it has a few flaws that prevented me from enjoying it more. I received an eARC from NetGalley and publisher Basic Books in exchange for a review.

Beginning with ancient mathematics, Proof is basically a subset of epistemology writ large, seeking to answer not “how do we know what we know?” but “how do we know what we know is what is?” But it doesn’t stop with math: Kucharski takes us from math to law, medicine, and society at large. This is what makes the book so fascinating and relevant, I think. Given the rise of misinformation or junk data, which Proof touches on, it’s more important now than ever to be able to evaluate what we should or should not believe—and to remember we can’t always trust our senses.

While there is plenty in this book I knew already, I also learned a lot. Kucharski has a great talent for diving deep into a subject and then only extracting the parts he wants to show the reader, like a diver who knows exactly how to pluck the best pearl or lobster and bring it up to the surface for onlookers. He doesn’t make you go down the rabbit hole with him, and that’s awesome. Proof is quite readable in that way despite its interdisciplinary nature.

Where it stumbles, in my opinion, is honestly the editing. As a copyeditor (mind you, that’s different from structural, or developmental, editing), I’m loath to criticize my own, but … Kucharski, my guy, someone needed to tell you no more often. Or maybe you needed to listen. See, Proof’s chapters are long. Like, I would hit a new chapter (and this is one of those rare eARCs that were actually formatted as ebooks rather than converted PDFs, woohoo, so my ereader detected chapters), and my ereader would be like, “This chapter will take twenty minutes to read,” and I was like, “Um, what?” because, look, I am no slouch in the speed department, so my ereader knows how fast I read. Sure enough, however … yeah, those chapters are long. Because Kucharski likes to digress, likes to ramble, likes to pack in as much as he can, and it seems like not a single person cared enough to stop him! Moreover, the chapters lack sProof: The Uncertain Science of Certaintytrong organizational structures, such as stories linked to individuals or any sense of narrative.

The result is a paradox of a book: the information is interesting, yet the writing is dull and ponderous. It’s a great example of why science and science communication are two different fields of study, and why being good at one doesn’t make you good at the other. Kucharski clearly knows his stuff, yet his storytelling skills aren’t a match, and it shows.

Proof is fascinating in its own way, and you would probably learn a lot if you picked up this book. But it’s not going to be one that sticks around for me.

Proof is a (mostly) layman accessible monograph on scientific certainty and methods, written by Dr. Adam Kucharski. Released 6th May 2025 by Hachette on their Basic Books imprint, it's 368 pages and is available in hardcover, audio, and ebook formats. It's worth noting that the ebook format has a handy interactive table of contents as well as interactive links and references throughout.

The author has a wide ranging background: from mathematics to infectious diseases/tropical medicine/epidemiology. This book is similarly wide ranging and not always direct - there's a fair bit of meandering involved. That being said, the author writes well and understandably and doesn't rely on obfuscation and jargon to score points.

Although not academically rigorous, the content is meticulously annotated throughout, and the chapter notes are likely worth the price of the book for the extra content alone. As a multidisciplinary work, there are some wide ranging meanders across disciplines and subjects (and time periods), but overall, it's interesting and fairly engaging. The author is clearly bright, and invested in the subject, and it's indisputably important and noble, helping humans understand how to winnow *actual* facts from the chaff we're all bombarded with continually.

Three and a half stars. It would be a good choice for public or post-secondary school library acquisition, or for niche non-fiction readers.

Disclosure: I received an ARC at no cost from the author/publisher for review purposes.

Thank you, NatGalley for providing an ARC of this book.

If you never read a book about statistics and maths, you will find this one quite amusing. If you have, then there is likely not enough new things for you to render this a definitive buy.

Let me preface my review by the fact that I’m a scientist and have a quantitative background.

This probably meant that I was expecting something different. A book by a mathematician talking about different types of proofs and logical arguments which inevitably lead to the uncertainty of using data to make arguments.

What the book actually was: largely unconnected collections of thoughts and stories about how we sort of proof things. I found the writing unfocused and meandering (sometimes with an unexpected abruptness the author pivots to a different topic).

As a I stated before, because it’s in my line of work and because I read books on statistics, a good chunk of this book was just a repetition and notes on concepts I know probably too much about. I also found the explanations of these statistical concepts not clear enough (especially the p-value). So for anyone looking to learn about these, you are better off with David Spiegelhalter.

Not As Bad As It Could Have Been. Quite honestly, if I had known up front that I was reading a book about proof and certainty written by a *COVID "scientist"*... I would never have picked the damn book up to begin with. Those fuckers have been more wrong than flat earthers, and the world learned from dire direct personal experience to not believe a word they say.

This noted, Kucharski does at least admit that even he was wrong in at least certain areas, so the fact that he wasn't trying to defend everything he and his colleagues did to us and all of their blatant mistakes was at least somewhat refreshing and gave this review its title.

Kucharski actually does a good job here with writing about precise concepts in layman and approachable terms, and even raises great (and hitherto unknown even to me) points about how even Euclid's Principia ultimately shaped decades of American politics... via one State Senator working himself through it in order to learn how to be a more convincing orator. That particular State Senator being none other than later President of the United States Abraham Lincoln.

Similarly, other sections are also quite enlightening about other forms of proof, even going so far as to at least allow for the possibility of Bayes being wrong in his Theorem (a topic explored much more fully in the much more targeted work Bernoulli's Fallacy by Aubrey Clayton) - which not many books about proof and statistics have ever done, at least in my experience as a former math and math related fields major and avid reader. (LONG story short, I came within a half dozen classes of getting degrees in Computer Science (the one I ultimately did get), Mathematics, and Secondary Mathematics Education at once... almost 20 yrs ago to the day as I write this review.)

Ultimately, the star deductions are for the long focus on COVID, which even 5 yrs later still warrants a star deduction in my own personal war against books focusing on that topic, and for trying to defend the scientists who were pushing so much of the damaging narratives - including, it seems, Kucharski himself. In a book about "proof" and "certainty", where history has now proven that one group of scientists in particular was so *incredibly* wrong in their "certain" judgements, to defend that very group of scientists as correct is to actively deny reality, and this cannot be ignored in such a text as this. (I've since come to forgive/ be far more lenient about more passing references to that horrible period of the 21st century, by the way.)

Now, maybe your political positions align more with Kucharski's. Maybe you still believe the blatant lies the world was fed about that period that ultimately caused far more harm and devastation than the actual virus ever did. In which case, you're going to LOVE this book.

But for those like me who believe that every single one of those "scientists" should find a more appropriate job that suits their actual knowledge and skill level - burger flipper, maybe? - eh... read this book anyway. Kucharski really does have some great stuff here, when he's not talking COVID.

Recommended.

Adam Kucharski’s latest book, Proof: The Uncertain Science of Certainty, offers a compelling exploration of how we define, assess, and communicate truth in an increasingly complex and information-saturated world. Building upon his expertise in mathematics and epidemiology, Kucharski examines the limitations of traditional notions of proof and the challenges of establishing certainty across various domains, including science, law, and public policy.​

At the heart of Proof is the assertion that the concept of proof is far more nuanced than often assumed. Kucharski illustrates this by referencing the Monty Hall problem, a probability puzzle that even confounded renowned mathematician Paul Erdős, to demonstrate how human intuition can falter in the face of statistical reasoning . He further delves into historical shifts in mathematical understanding, such as the development of non-Euclidean geometry and Gödel's incompleteness theorems, to highlight the evolving nature of what constitutes proof.

The book also scrutinizes the application of algorithms in areas like criminal justice, where tools like the Correctional Offender Management Profiling for Alternative Sanctions (COMPAS) have been shown to exhibit biases, raising questions about fairness and the reliability of algorithmic decision-making. In the realm of medicine, Kucharski challenges the supremacy of randomized controlled trials (RCTs) as the "gold standard" for evidence, arguing that other forms of proof, including experiential knowledge, can be equally valid depending on the context. I'm personally a bit skeptical of this approach, but don't think RCTs are the only source of evidence worth reviewing.​

Kucharski supports his arguments with an onslaught of historical anecdotes, case studies, and analyses of current events. For instance, he recounts how Abraham Lincoln employed Euclidean logic to argue against slavery, illustrating the intersection of mathematical reasoning and moral philosophy. He also discusses the origins of clinical trial designs, tracing them back to informal discussions at Rothamsted agricultural station, thereby emphasizing the social dimensions of scientific development .​

The book's strength lies in its ability to make complex concepts accessible without oversimplification. Kucharski's clear prose and engaging storytelling render intricate ideas in mathematics and science approachable for general readers. His balanced examination of topics, acknowledging both the utility and limitations of various forms of proof, enhances the book's credibility and relevance.​

Proof is particularly timely, addressing the pervasive issues of misinformation and the erosion of public trust in expert knowledge. By dissecting the processes through which we arrive at conclusions and the potential pitfalls therein, Kucharski equips readers with a critical framework for evaluating information. His call for humility and transparency in the communication of scientific findings resonates strongly in an era marked by polarized discourse and skepticism toward established institutions.​ I always respect epistemic humility.

While the book addresses sophisticated topics, its lucid explanations and relatable examples make it accessible to a broad audience. Readers without a background in mathematics or science will find the discussions enlightening rather than daunting.​ I strongly recommend the book.

“Proof,” by Adam Kucharski, is a thought-provoking read about how we obtain comfort and reach consensus that something is true in the absence of definitive proof. The chapters present situations (e.g., legal verdicts, medical testing, artificial intelligence) where this occurs and cover the mathematical and scientific approaches used to procure ample proof in these situations. The chapters also provide a history of how adequate proof has been attained throughout time, from Euclidean definitions and self-evident axioms (circa 300 BC) to Newtonian physics to modern-day AI models.

The book is nonmathematical in nature. However, the chapters do discuss many statistical terms and methods (at a high-level) to provide context and clarity to the situations considered. Some included terms and methods are: null hypothesis, p-values, confidence intervals, randomized controlled trials (RCTs), Type I and Type II errors, and various proof types (e.g., by exhaustion, by simulation, by contradiction, etc.).

The legal standards of “preponderance of the evidence” (civil cases) and “proof beyond a reasonable doubt” (criminal cases) are also discussed, as well as their inherent problems in balancing wrongful convictions against freeing the guilty. The author, who was an epidemiologist during the COVID-19 pandemic, also shares his experiences and insights throughout the pandemic—including difficulties related to data collection, variants, testing, vaccines, public sentiment, media coverage, politics, and obtaining consensus.

Some lighter proof situations are also discussed. One sets out to establish why tea tastes better when milk is poured into a cup before tea is added vs. afterward. Another involves the transition that Guiness made from traditional beer-making (dependent on human judgment) to industrial brewing, and the science and methods used to do so without diminishing beer quality.

Throughout the book, the author adequately demonstrates why proof is difficult to obtain. Even when sufficient evidence is available to convince ourselves that something is correct, the next step is to convince others—which can be particularly difficult due to factors such as psychology, politics, and prior beliefs. The author suggests that we continually learn to embrace uncertainty, balance our beliefs, and update our tools and approaches to modern problems that cannot be solved using old methods. The author also proposes that we increasingly trust our researchers, institutions, etc., while being cognizant of disinformation and other tactics used to distort the truth in modern society.

“Proof” is an excellent read that opens your mind and allows you to better recognize situations that lack certain truth. It also helps you more fully appreciate the difficulties that researchers and institutions encounter, and the complex decisions that must be made, when definitive proof is unobtainable.

[My special thanks to Basic Books (Hachette Book Group) and NetGalley for an advance reader’s copy of this book.]

I enjoyed the conversational tone of the writing and the good pacing. But I did find the explanations overly complex. I also felt that there was too much on artificial intelligence and was uncertain how this fit with the rest of the book. I think that people with greater background will enjoy the book more than I did. Thank you to Netgalley and Basic Books for the advance reader copy.

Thank you to NetGalley, the author, and the publisher for this complementary ARC in exchange for my honest review!

This book is a sort of hybrid math-science-history book, covering all sorts of different topics revolving around how we prove things. Each chapter touches on a different subject and goes in depth about how proof factors in - I found the legal section particularly interesting.

I had hoped that the math portion had gone a bit more in depth but overall I would recommend this to anyone looking for a more broad perspective of how things are proved.

Ideas like evidence and proof are at the heart of so many aspects of modern life. On the surface they seem to be relatively simple and clear concepts, which anyone can use. But the author shows that there are surprising levels of underlying complexity, to both the interpretation of what proof means and to how it is applied in real world situations.

With chapters exploring proof in contexts of politics, mathematics, law, medicine and daily life, the book presents a wealth of examples in each chapter, to illustrate how and why proof is applied in the way that it typically is. Whether it be calculating the numbers of enemy tanks on a battlefield, or matching teeth to bite marks, the author shows that there have been remarkable developments (and failures) in what can be considered to be proof.

The chapter on legal issues was particularly interesting. It referred to cases from a wide range of jurisdictions, which included subtly different legal principles and some very nuanced models of proof. In one country the bar for conviction was ‘high,’ so that there was less risk of accidentally convicting innocent people. In another country the bar was ‘low’ so that it didn’t matter if innocent people were accidentally convicted. The difference between a ‘high’ and a ‘low’ bar often revolved around the principles and methodology of what counted as proof.

One of the issues often in the background of proof, is the question of how key words are interpreted. This was illustrated well with a number of historical examples, such as segregated education. At the beginning of the twentieth century segregated education was understood to be compatible with equality. By the middle of the twentieth century that was no longer the case. Changing social understandings can completely reverse what counts as a proof in different eras.

Where I think that the book could have pressed issues a little harder, was with the factors surrounding proof in the context of public policy. There are some very real contemporary questions about the competing policies of political parties. If the red party argues for policy x, and the blue party argues for policy y, what does it mean for one policy to be proved ‘better’ than another? Is it simply a matter of votes and preferences, or are there other factors which are relevant, and which perhaps drive votes and preferences?

Overall, this was an informative book which should be accessible to readers from any background. In places. the complexities of the issues mean that the book will probably be most appreciated by graduate readers. Around 20% of the text consists of notes, so there are links and follow up references for readers who want to pursue matters further.

I should add that this is a review of a (free) digital Advanced Review Copy (ARC) of the text. It carried a note to reviewers which implied that there may be differences between the text which I saw, and the final published copy. If that is so, then that means that elements of the rating and comments above may not accurately reflect what the final version of the book ends up looking like.