Unknowability

In philosophy, unknowability is a component of philosophical inquiry that explores the possibility of inherently unaccessible knowledge. It addresses aspects of epistemology of that which we cannot know. Baruch Spinoza's Theory of Attributes [1] argues that a human's finite mind cannot apprehend infinite substance, and accordingly, infinite substance as it is in itself is in-principal unknowable to the finite mind. Immanuel Kant brought focus to unknowability theory in his use of the noumenon concept postulating that while we can know the noumenal exists, it is not itself sensible and must therefore remain unknowable.

Modern inquiry encompasses questions such as the Halting Problem which in their very nature cannot be possibly answered. This area of study has a long and somewhat diffuse history as the challenge arises in many areas of scholarly and practical investigations.

Nicholas Rescher has provided the most recent focused scholarship for this area in Unknowability: An Inquiry into the Limits of Knowledge[2], where he offered three high level categories, logical unknowability, conceptual unknowability, and in-principle unknowability.

Rescher's categories of unknowability

Rescher organizes unknowability in three major categories:

  • logical unknowability — arising from abstract considerations of epistemic logic.
  • conceptual unknowability — analytically demonstrable of unknowability based on concepts and involved.
  • in-principle unknowability — based on fundamental principles.

In-principle unknowability may also be due to a need for more energy and matter than is available in the universe to answer a question, or due to fundamental reasons associated with the quantum nature of matter. In physics of special and general relativity, the Light cone marks the boundary of physically knowable events[3] [4]

The halting problem

The Halting Problem is a prominent example of an unknowability associated with established mathematical and logical specialties of computability theory, namely the problem of determining if arbitrary computer programs will ever finish running demonstrate in-principal unknowability. The current assessment is that halting problem is undecidable. There are other similarly significant questions provided in the list of undecidable problems.

It is sometimes argued that in principle, many problems can be reduced to the Halting Problem, for example the Diophantine problem, a decision problem that asks whether a given polynomial equation with integer coefficients has a solution in integers.

In 1936, Alan Turing proved that the halting problem is undecidable. This means that there is no algorithm that can take as input a program and determine whether it will halt. In 1970, Yuri Matiyasevich proved that the Diophantine problem (closely related to Hilbert's tenth problem) is also undecidable [5]. This means that there is no algorithm that can take as input a Diophantine equation and determine whether it has a solution in integers.

The relationship between the halting problem and the Diophantine problem is that the Diophantine problem can be reduced to the halting problem. This means that if there were an algorithm that could solve the Diophantine problem, then there would also be an algorithm that could solve the halting problem. However, since the halting problem is undecidable, this means that the Diophantine problem must also be undecidable.

The undecidability of the halting problem and the Diophantine problem has a number of implications for mathematics and computer science. For example, it means that there is no general algorithm for proving that a given mathematical statement is true or false. It also means that there is no general algorithm for finding solutions to Diophantine equations.

Gödel's incompleteness theorems demonstrate implicit in-principal unknowability of methods to prove consistency and completeness of foundation mathematical systems.

Graduations of Unknowability: There are systemic unknowabilities associated with frameworks of discussion. For example unknowability to particular individual humans (due to individual limitations), unknowability to humans at a particular time (due to lack of appropriate tools), unknowability to humans due to limits of matter and energy in the universe that might be required to conduct the appropriate experiments or conduct the calculations required. Unknowability to any processes, organism, or artifact.


Treatment of knowledge has been wide and diverse. Wikipedia itself is an initiate to capture and record knowledge using contemporary technological tools. Earlier attempts to capture and record knowledge include writing deep tracts on specific topics as well as the use of encyclopedias to organize and summarize entire fields or event the entirety of human knowledge.

Limits of knowledge

An associated topic that comes up frequently is that of Limits of Knowledge. A check on the number of titles in World Catalog containing 'limits of knowledge' in their title on 8 May 2023 yields over 800 results. results[6].

Examples of scholarly discussions limits of knowledge include:

  • John Horgan's End of science : facing the limits of knowledge in the twilight of the scientific age.[7]
  • Tavel Morton's Contemporary physics and the limits of knowledge. [8]
  • Ignoramus et ignorabimus - The Latin maxim meaning "we do not know and will not know", popularized by Emil du Bois-Reymond. Bois-Reymond's ignorabimus proclamation was viewed by David Hilbert as unsatisfactory, and motivated Hilbert to declare in 1900 International Congress of Mathematicians that answers to problems of mathematics are possible with human effort. He declared, "in mathematics there is no ignorabimus",[10]. The Halting Problem and the Diophantine Problem eventually were answered demonstrating in-principle unknowability of answers to some foundational mathematical questions (meaning ignorabimus Bois-Reymond's assertion was in fact correct.

Gregory Chaitin discusses unknowability in many of his works.

Categories of unknowns

Popular discussion of unknowability grew with the use of the phrase There are unknown unknowns by United States Secretary of Defense Donald Rumsfeld at a news briefing on February 12, 2002. In addition to unknown unknowns there are known unknowns and unknown knowns. These category labels appeared in discussion of identification of chemical substances[11][12][13].

Chaos theory

Chaos theory is a theory of dynamics that argues that even if we know initial conditions fairly well, measurements errors and computational limitation render fully correct long-term prediction impossible, hence guaranteeing ultimate unknowability of physical system behaviors.

References
  1. "Spinoza's Theory of Attributes". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. 2018.
  2. Rescher, Nicholas. Unknowability: an inquiry into the limits of knowledge. Lexington Books, 2009. https://www.worldcat.org/title/298538038
  3. Hilary Putnam, Time and Physical Geometry, The Journal of Philosophy, Vol. 64, No. 8 (Apr. 27, 1967), pp. 240-247 https://www.jstor.org/stable/2024493 https://doi.org/10.2307/2024493
  4. John M. Myers, F. Hadi Madjid, "Logical synchronization: how evidence and hypotheses steer atomic clocks," Proc. SPIE 9123, Quantum Information and Computation XII, 91230T (22 May 2014); https://doi.org/10.1117/12.2054945
  5. Matii︠a︡sevich I︠U︡. V. Hilbert's Tenth Problem. MIT Press 1993.https://www.worldcat.org/title/28424180
  6. https://www.worldcat.org/search?q=ti%3A%22limits+of+knowledge%22
  7. Horgan, John. The End of Science : Facing the Limits of Knowledge in the Twilight of the Scientific Age. Addison-Wesley Pub 1996. https://www.worldcat.org/title/34076685
  8. Tavel, Morton. Contemporary Physics and the Limits of Knowledge. Rutgers University Press 2002. https://www.worldcat.org/title/47838409
  9. Cherniak, Christopher. "Limits for knowledge." Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition 49.1 (1986): 1-18.https://www.jstor.org/stable/4319805
  10. Hilbert, David (1902). "Mathematical Problems: Lecture Delivered before the International Congress of Mathematicians at Paris in 1900". Bulletin of the American Mathematical Society. 8: 437–79. doi:10.1090/S0002-9904-1902-00923-3. MR 1557926.
  11. Little, James L. (2011). "Identification of "known unknowns" utilizing accurate mass data and ChemSpider". Journal of the American Society for Mass Spectrometry. 23 (1): 179–185. doi:10.1007/s13361-011-0265-y. PMID 22069037.
  12. McEachran, Andrew D.; Sobus, Jon R.; Williams, Antony J. (2016). "Identifying known unknowns using the US EPA's CompTox Chemistry Dashboard". Analytical and Bioanalytical Chemistry. 409 (7): 1729–1735. doi:10.1007/s00216-016-0139-z. PMID 27987027. S2CID 31754962.
  13. Schymanski, Emma L.; Williams, Antony J. (2017). "Open Science for Identifying "Known Unknown" Chemicals". Environmental Science and Technology. 51 (10): 5357–5359. Bibcode:2017EnST...51.5357S. doi:10.1021/acs.est.7b01908. PMC 6260822. PMID 28475325.

Further reading
  • Chaitin, Gregory J. The unknowable. Springer Science & Business Media, 1999. https://www.worldcat.org/title/41273107
  • DeNicola, Daniel R. Understanding ignorance: The surprising impact of what we don't know. MIT Press, 2017
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