Gödel's Incompleteness Theorem

Kurt Gödel was an Austrian-American logician and mathematician who was born in 1906 in the city of Brünn, in Austria-Hungary (now the city of Brno, in the Czech Republic). He studied and taught mathematics at the University of Vienna, where he participated in meetings of the Vienna Circle, the famous group of philosophers that included Moritz Schlick (1882-1936) and Rudolf Carnap (1891-1970). In 1931, at the age of 25, he published a proof that is now known as Gödel's Incompleteness Theorem and that demonstrated the existence of formally undecidable propositions in any formal system of arithmetic (such as that described by Alfred North Whitehead and Bertrand Russell in the Principia Mathematica).¹ This theorem is one of the most famous theorems in modern mathematics.² In 1939, Gödel and his wife Adele emigrated to the United States, where he became a member of the faculty of the Institute for Advanced Study in Princeton, N.J., and where he developed a close friendship with Albert Einstein. Gödel and his wife became U.S. citizens in 1948. He died in Princeton, N.J. in 1978.

      One way of roughly summarizing Gödel's (First) Incompleteness Theorem, as described by the cognitive scientist Douglas Hofstadter (1999), is to say that it shows that any consistent axiomatic formulation of number theory includes undecidable propositions (propositions that can neither be proven nor be disproven within that axiomatic formulation).³

      Another way of roughly summarizing Gödel's (First) Incompleteness Theorem, as described by the logician and mathematician Jean van Heijenoort (1967), is to say that it shows that in any formal system adequate for number theory, there are formulas that are neither provable nor disprovable. A corollary to this theorem is that the consistency of any formal system adequate for number theory cannot be proven within that system.⁴ No consistent formal system that is adequate for number theory can prove its own consistency. This is known as Gödel's Second Incompleteness Theorem.

      Another way of roughly summarizing Gödel's (First) Incompleteness Theorem, as described by the logician and philosopher Geoffrey Hunter (1971), is to say that it shows that for any consistent system S of formal arithmetic, if S has decidable sets of formulas and proofs and contains representations of every decidable set of natural numbers, then S is incomplete.⁵ Thus, for any consistent system of formal arithmetic in which (1) the set of axioms and the rules of inference are recursively definable, and (2) every recursive relation is definable, there are undecidable arithmetical propositions of the form (x)Fx, where F is a recursively defined property of natural numbers.⁶ 

      Another way of roughly summarizing Gödel's (First) Incompleteness Theorem, as described by the logician and philosopher Peter Suber (2002), is to say that it shows that any consistent formal system of arithmetic that is of sufficient strength (to have decidable sets of formulas and proofs and to represent every decidable set of natural numbers) is deductively incomplete.⁷ In other words, any consistent formal system of arithmetic that is of sufficient strength contains arithmetical propositions that are undecidable (that are neither provable nor disprovable within that system). 

      This means that within any consistent nontrivial formal system of arithmetic, there are arithmetical propositions that are true or false but that cannot be proven to be true or false within that system. The axioms and inference rules of any consistent nontrivial formal system of arithmetic are insufficient to decide the truth or falsehood of every arithmetical proposition expressible within that system. It is impossible to devise a consistent nontrivial system of formal arithmetic whose axioms and inference rules are complete enough to prove or disprove all the arithmetical propositions expressible within that system. No formal system of arithmetic that is of sufficient strength to have decidable sets of formulas and proofs and to represent every decidable set of natural numbers can be both consistent and deductively complete.⁸

      Moreover, a consistent formal system adequate for number theory cannot prove its own consistency. It cannot guarantee that it will not contain some inconsistency.

      This means that in order for every possible proposition expressible within a nontrivial formal system of arithmetic to be provable or disprovable, the system has to be in some way inconsistent. Deductive completeness within such a system must therefore come at the price of inconsistency. All consistent nontrivial formal systems of arithmetic are deductively incomplete.

      The logician and philosopher Jaako Hintikka (1996) explains that it is important not to confuse the incompleteness of a nonlogical (mathematical) theory with the nonaxiomatizability of a logical theory.⁹ He also explains that it is important not to confuse deductive incompleteness (the inability of an axiomatic system to prove or disprove all propositions expressible within that system) with semantic incompleteness (the inability of an axiomatic system to express as theorems all logically valid sentences of the underlying language), or with descriptive incompleteness (the inability of a formal system to provide models of all the objects or sets of objects that it is asked to describe), or with Hilbertian incompleteness (the inability of an axiomatic system to provide a set of axiomatic models to which none can be added without violating the axioms of that system).

¹Larousse Biographical Dictionary, ed. by Magnus Magnusson and Rosemary Goring (New York: Larousse, 1990), p. 598.

²Ibid., p. 598.

³Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Goldren Braid (New York: Basic Books, 1999), p. 17.

⁴Jean van Heijenoort, "Gödel's Theorem," in The Encyclopedia of Philosophy, ed. by Paul Edwards, Vol. 3 (New York: MacMillan, 1967), p. 348. 

⁵Geoffrey Hunter, Metalogic: An Introduction to the Metatheory of Standard First Order Logic (Berkeley: University of California Press, 1971), p. 228.

⁶Kurt Gödel, "On formally undecidable propositions of Principia Mathematica and related systems" [Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme," 1931] in Kurt Gödel Collected Works, Volume I, edited by Solomon Feferman, et al. (Oxford: Oxford University Press, 2004) p. 181.

⁷Peter Suber, "Glossary of First-Order Logic," (1999-2002), online at http://www.earlham.edu/~peters/courses/logsys/glossary.htm.

⁸Ibid.

⁹Jaako Hintikka, The Principles of Mathematics Revisited (Cambridge: Cambridge University Press, 1996), p. 91.

Leibniz's Law

G.W. Leibniz (1646-1716) says in Section IX of his Discourse on Metaphysics (Discours de Métaphysique, 1686) that no two substances can be exactly alike. This is known as Leibniz's Law. Another way of expressing this is: No two substances can be exactly the same and yet be numerically different. If two substances were exactly the same, then they would be the same substance and would not be two separate substances. 

      Leibniz's Law (that no two things can share all their properties in common) can be expressed in a positive way as follows: if two things are identical, then they share all their properties in common (this metaphysical principle is called the indiscernibility of identicals), and conversely, if two things share all their properties in common, then they are identical (this metaphysical principle is called the identity of indiscernibles). According to the indiscernibility of identicals, if two things are identical, then no difference between them is discernible, and according to the identity of indiscernibles, if no difference is discernible between two things, then they are identical.

      An objection that might be raised to the identity of indiscernibles is that if two things are superimposed on each other, then they might temporarily share all their properties in common and yet not be the same. It might not be possible to discern that they are two things, rather than one. An objection that might be raised to the indiscernibility of identicals is that if two things are, to any means of detection, identical, then they might still differ from each other in a property that is undetectable. It might not be possible in practice to detect any difference between them. 

      Thus, a distinction may need to be made between theoretical and practical discernibility. The difference between two nonidentical things may in some cases not be practically discernible or verifiable. Similarly, the sameness of two things may in some cases not be practically discernible or verifiable.             

      Leibniz's Law can be expressed symbolically as (x)(y) [x=y → (F)(Fx ↔ Fy)], which may be read as "for every x and for every y, if x is identical to y, then every property F that is possessed by x is also possessed by y, and every property F that is possessed by y is also possessed by x" (this is the indiscernibility of identicals), and conversely as (x)(y) [(F )(Fx ↔ Fy) → x=y], which may be read as "for every x and for every y, if every property F that is possessed by x is also possessed by y, and every property F that is possessed by y is also possessed by x, then x is identical to y" (this is the identity of indiscernibles).
      The philosopher Max Black (1952) offers several arguments against the principle that if no difference is discernible between two things, then they are identical. He argues that two things cannot be identical, since if they were, then they would be only one thing, and not two. If we say that a is identical to b, then we are merely using two different names to refer to the same thing. And if a and b are merely two different names for the same thing, then when we say that "a is identical to b," we are merely saying that "a is a," which is a tautology. The principle that "If there is no difference between a and b, then they are the same" is trivial. And if there were a universe consisting of two exactly similar spheres, then conceivably two things could share the same properties and still not be the same, and thus the identity of indiscernibles would again be put into question.¹

      It may, however, be worth noting that two things may be similar to, or the same as, each other in possessing many distinct kinds of properties. Identity between two things may involve material, formal, spatial, temporal, relational, and other kinds of properties.

      Can an exact duplicate or replica of something be properly called "identical to" or "the same as" that thing? If so, why may there still be some doubt or uncertainty about whether the two things are alike in every respect? What may happen to the identity of the two things as they change over a period of time?

      Do changes in the properties of things always change the natures of those things? Moreover, do changes in the physical, intellectual, emotional, or social attributes of a person always change the nature of that person?  Are you the same person that you were 5 minutes ago? 7 days ago? 5 years ago?
      Surely, there must be some properties that are relevant to sameness, and some that are irrelevant. Should we then relativize or qualify the indiscernibility of identicals by saying that in order for two things to be the same, they must share all properties that are essential or relevant to their sameness? Some properties may be essential to the identity of two things, while other properties may be unessential. 

¹Max Black, "The Identity of Indiscernibles," in Mind, Vol. 61, No. 242 (April, 1952) pp. 153-164.

Moral Agency and Personhood

In an article published in the February 18, 2010 edition of The New England Journal of Medicine, Martin Monti et al. reported that functional MRI scanning may detect awareness and cognition in some patients with severe brain injuries who have no other evidence of sustained, reproducible, purposeful, or voluntary behavioral responses to visual, auditory, tactile, or noxious stimuli. Thus, a small proportion of such patients who have severe brain damage and who appear to be unconscious or only minimally conscious (with wakefulness but without awareness) may actually have some level of awareness as detected on functional MRI scanning, and may in fact be able to communicate by willfully modulating their brain activity.¹ 

      In an editorial in the same edition of the NEJM, Allan H. Ropper noted that "The unfortunate term 'vegetative' has been used to describe patients whose eyes open after a period of coma but who lack any meaningful responses to stimuli. Open eyes give the impression of normal alertness, but the patient's behavioral repertoire is limited to reflexive actions such as posturing or purposeless movements, roving eye movements, swallowing, and yawning."² 

      The term "persistent vegetative state" (PVS) has been used to refer to prolonged states of unconsciousness (due to traumatic or non-traumatic brain injuries, degenerative or metabolic brain disorders, or severe congenital malformations of the nervous system) in which the patient is completely unaware of self and of the environment but has sleep-wake cycles, with preservation of hypothalamic and brainstem autonomic functions. Such patients "show no evidence of sustained, reproducible, purposeful or voluntary behavioral responses to visual, auditory, tactile, or noxious stimuli; show no evidence of language comprehension or expression; have bowel and bladder incontinence; and have variably preserved cranial nerve and spinal reflexes."³ They may be able to exhibit such behaviors as blinking, swallowing, groaning, grimacing, breathing spontaneously, and reflex posturing of the limbs, but they are unable to eat, drink, talk, or make purposeful limb movements. As a result of being permanently confined to bed or chair, they also tend to develop such complications as muscle wasting, limb and joint contractures, skin breakdown, pressure sores, recurrent urinary tract infections, and pneumonia.⁴

      However, the term "vegetative" has pejorative connotations, and implies that the patient who is in this state of unconsciousness is merely vegetating or is merely a vegetable. The term diminishes the personhood of the person who is in this condition.

      The study by Monti et al. in the Feb. 18, 2010 edition of the NEJM demonstrates that a small proportion of patients who have been previously diagnosed as having entered a "persistent vegetative state" may actually have some awareness and cognition. They may be able to respond to auditory stimuli, and they may be able to voluntarily modify their brain activity in order to perform simple communicational tasks.

      Are individuals with depressed levels of consciousness who are minimally aware of their environment but unable to make any behavioral responses able to retain their moral agency? Are individuals who are fully aware of their environment but unable to make any behavioral responses (such as completely paralyzed individuals, or individuals with the "locked-in syndrome," as originally described by Plum and Posner in 1966)⁵ able to retain their moral agency? Such individuals obviously retain their personhood, but can they still function as moral agents? Are they still responsible for whatever thoughts and feelings they may have? 

      What are the criteria for moral agency? Moral agency is a contributor to, but not a requirement for, personhood, since some persons (such as small children and the mentally impaired or handicapped) may not have sufficient voluntary control over their actions to be held morally responsible for them. The proper definition of personhood is quite controversial, since legal recognition of personhood conveys various rights and responsibilities upon any being who is legally considered to be a person. Thus, the politics of personhood has complicated the development of consensus regarding such issues as the nature of reproductive health rights, the legitimacy of end-of-life decision making, the nature of corporate personhood, and the nature of animal rights. 

      Any attempt to establish a definite set of criteria for personhood may be problematic, insofar as any such set of criteria may not be sufficiently inclusive. Every human being is a person. Whether non-humans can also be considered persons is a matter that has been much discussed elsewhere (e.g. Peter Singer, Practical Ethics, 1993; Sarah Chan and John Harris, "Human Animals and Nonhuman Persons," in The Oxford Handbook of Animal Ethics, ed. by Tom L. Beauchamp and R.G. Frey, 2011). Every person has a personality, i.e. a set of physical, cognitive, perceptual, mental, emotional, and social traits or characteristics that define him/her as a distinct individual. The identity of each person is also defined by his/her relations with other persons.

      Agency may be defined as the state of acting or of having the capacity to act (independently or in cooperation with other agents). An agent may be defined as an individual or collective entity that acts or is capable of acting. An agent may act autonomously or in cooperation with other agents--in, upon, through, because of, by means of, for the purpose of, or in behalf of some other agent or entity. 

      Moral agency may be defined as the state of acting or of having the capacity to act, such that the agent is in a position to be held morally responsible for their actions. In order to be held fully responsible for their actions, an agent must have sufficient autonomy (or freedom of choice) to be held fully responsible for their actions, their actions must be voluntary and under their own control, and the actions must be intentional.

     There may be many kinds of agency, including physical, moral, human, non-human, rational, legal, and professional agency. Physical agents include biological, chemical, environmental, radiologic, and thermal agents. Legal agents include law enforcement officers, attorneys, legal guardians, legal representatives, executors, contractors, brokers, and lobbyists. Business and professional agents include literary, theatrical, sports, travel, insurance, booking, sales, and real estate agents.
      Agents acting jointly or collectively may share a moral responsibility for their joint or collective actions. The nature of the responsibility of each agent for a joint or collective action may depend on whether each agent is a willing participant in the action, whether each agent is aware of the possible consequences of the action, and whether each agent has the ability to promote, prevent, or modify the expected or actual outcome of the action.

¹Martin M. Monti, et al. "Willful Modulation of Brain Activity in Disorders of Consciousness." N Eng J Med; 362: 579-589, Feb. 18, 2010.

²Allan H. Ropper. "Cogito Ergo Sum by MRI." N Engl J Med; 362: 648-649, Feb. 18, 2010.

³The Multi-Society Task Force on PVS. "Medical Aspects of the Persistent Vegetative State." N Engl J Med; 330: 1499-1508, May 26, 1994.
⁴American Hospice Foundation. "Coma and Persistent Vegetative State: An Exploration of Terms." 2005, at http://www.americanhospice.org/articles-mainmenu-8/caregiving-mainmenu-10/50-coma-and-persistent-vegetative-state-an-exploration-of-terms.

⁵Fred Plum and Jerome B. Posner. The Diagnosis of Stupor and Coma. Philadelphia: F.A. Davis and Company, 1966.

The Physician Philosopher

Describing oneself as a physician may be something that is more easily understandable to most people than describing oneself as a philosopher. If one were to list one's occupation on a job application as "physician," for example, most people would have a much clearer understanding of what one meant than if one were to list one's occupation as "philosopher." However, if the meaning of the term "philosopher" is restricted to "professional philosopher," i.e. to an individual who has an advanced degree (such as a Ph.D.) in philosophy and who earns their living as a professional philosopher, teaching and writing in the field, then the term becomes somewhat clearer and more comprehensible. But what about those students, writers, thinkers, and other laypeople who engage in philosophy but are not professional philosophers? Can public philosophy be meaningfully engaged in only by professional philosophers?

      I'm a physician and also an aspiring philosopher. I suppose that my ambition to become a "physician philosopher" began a number of years ago when a friend graciously, and quite surprisingly, referred to me as such. I'll leave it to others to judge whether my aspirations are mere pretensions. My profession is that of physician. However, I feel a calling to both medicine and philosophy. Which is the higher calling? Is there a higher calling than that of healing the sick and helping the needy (as in the case of medicine)? Is there a higher calling than that of discovering the meaning of truth and justice (as in the case of philosophy)? Does the career pathway that I've chosen mean that I can never become a "real philosopher"?

      And what exactly is a "physician philosopher"? The question may be an important one, because medical decision-making may in some cases require a physician to be able to think philosophically as well as scientifically. Practice in philosophical thinking may to some degree enable an individual to become a better physician. Indeed, it may be argued that the philosophy of medicine is founded on the premise that philosophical thinking may be useful in analyzing the practice of medicine and the treatment of medical problems; thus, engagement in the philosophy of medicine can yield insights that will lead to improved practice and treatment.

      Perhaps the best answer that I can give to the question, "What is a physician philosopher?" is that a physician philosopher is a person whose vocation is to be both physician and philosopher, inspired by love of both vocations (medicine and philosophy). A physician philosopher is a physician who has qualified as (or been certified or recognized as) a philosopher by virtue of their work (writing, teaching, research) in the field. A physician philosopher may also be a philosopher who is engaged in the practice of medicine, or a physician who does philosophy in an academic setting, or a physician who specializes in the philosophy of medicine, biomedical ethics, the philosophy of neuroscience, the philosophy of psychiatry, or some other related field.

      How can one become a physician philosopher? Simply put, one can become a physician philosopher by obtaining academic degrees in both medicine and philosophy. Many schools offer dual degree programs in medicine and philosophy. Bioethics training may also be available in relevant fields, such as clinical bioethics, research bioethics, public health ethics, global bioethics, health care economics, and health care resource allocation. Job opportunities for physicians with training in clinical or research bioethics may include employment as a board member or ethics consultant for a hospital ethics committee, employment as a compliance officer for an institutional review board (IRB) overseeing scientific research, employment as an instructor in bioethics at a health care institution (hospital, school of public health, medical school, or nursing school), and employment as a faculty member (academic bioethicist) at a center for bioethics. Websites that offer information regarding careers in bioethics may be found at http://www.nature.com/nbt/journal/v22/n2/full/nbt0204-247.html, and at http://www.bioethics.net/jobs/.

      What kind of person should a physician philosopher be? Perhaps the best answer that I can give to this question is that the ideal physician philosopher is a person who has a (moral, intellectual, and social) commitment to, and engagement in, both vocations, and who does not view philosophy as a means to practice medicine or medicine as a means to engage in philosophy. An ideal physician philosopher is a person who has a love for medicine (a love for curing illness and alleviating suffering, not out of any sense of paternalism or superiority in relation to those in need of medical care, but out of a selfless need to help and care for others, and out of a sense of wanting to serve humanity and contribute to human well-being), and who has a love for philosophy (a love of wisdom, clarity, precision, intellectual discipline, and discovering the true nature of things). An ideal physician philosopher is a physician who is engaged in philosophy not merely for its medical applications, but for its wider applications to human knowledge, understanding, and well-being. 

      What role should a physician philosopher play in society? One of the most useful roles that a physician philosopher may play in society is that of questioning and clarifying unspoken assumptions about the nature of human life and existence that impact care for the sick, the helpless, and the needy. The physician philosopher may attempt to clarify and answer epistemological, ontological, ethical, linguistic, social, and political questions that influence our response to sickness and health. 

      Another role that the physician philosopher may play is that of clarifying the meaning of good health, disease prevention, distributive justice in health care, etc., and providing an analysis or explanation of how these goals may best be achieved.

      The following is a list (admittedly, very incomplete) of noted physician philosophers: Galen, Sextus Empiricus, Ibn Sina, Ibn Rushd, Al-Kindi, Moses Maimonides, John Locke, William James, Sigmund Freud, Karl Jung, Albert Schweitzer, Maria Montessori, Karl Jaspers, Victor Frankl, Frantz Fanon, Thomas Szasz, R.D. Laing, Walker Percy, Edmund Pellegrino, Nawal El Saadawi, Antonio Damasio, Nayef Al-Rodhan, Drew Leder, Leon Kass, Daniel Sulmasy, and Deepak Chopra.

      It should be noted that there may in some cases be a closer link between medicine and philosophy in Eastern society (e.g. China, Tibet, Korea) than in Western society. Traditional Chinese medicine, for example, may be more holistic in its approach than Western medicine and may place a greater emphasis on the relation between spiritual and physical healing. The role of the physician as philosopher (or at least as the vehicle of a particular philosophy) may in such cases be a potentially more integrated one than in cases (in the East and West) where there is a more scientific, technological approach to medicine. The latter approach may sometimes make it more difficult to define the proper relation between medicine and philosophy.

Medicine and Philosophy

What do medicine and philosophy have in common? Why should philosophy as a mode of inquiry be of interest to physicians? How can philosophy be a productive and rewarding endeavor for physicians? How is philosophy relevant to the practice of medicine? What can philosophy contribute to the practice of medicine?

      Medicine and philosophy may mutually support and enhance each other in a number of ways. Philosophy has traditionally included such fields as epistemology, metaphysics, ethics, aesthetics, logic, and political philosophy, and each of these disciplines may have practical applications to medicine.

      Epistemology may clarify the way in which health care providers (individual practitioners as well as group providers, administrators, and hospitals) form beliefs and opinions, make decisions, and respond to various sources of evidence. Social epistemology may also reveal the social dimensions of knowledge and information, and it may clarify the relevance of social institutions (such as scientific journals, hospital committees, professional licensing boards, and government agencies) to knowledge.

      Metaphysics and ontology may provide models of social reality that are useful in analyzing various kinds of interactions between physicians and patients, between patients and physicians and third parties, and between physicians and society.

      Ethics (particularly, biomedical ethics) is relevant to the practice of medicine, because it addresses such questions as: What ethical principles should guide the physician-patient relationship? Do physicians in private practice have a moral obligation to provide services to patients who cannot afford to pay? Do physicians entering private practice have a moral obligation to practice in geographic areas that are underserved with regard to the availability of health care services? Do physicians in private practice have a moral obligation to accept public health insurance payments as full reimbursement for services provided?

         Other ethical questions regarding health care that are of great concern to society include: How can equitability of access to health care best be achieved? Should an individual's access to health care be based on their medical need rather than their ability to pay? What constitute fair and just principles of health care resource allocation? What corrective measures should be taken to address health care inequities? What should be done to prevent wasteful and unnecessary overutilization of medical services? What should be done to ensure that services are appropriate, timely, and cost-effective? What responsibilities do wealthy nations have to address global health care needs? 

         Biomedical ethics is also an important field of study for physicians because it addresses issues of professional conduct, such as appropriate prescription writing, appropriate provision of services, adherence to reasonable standards of care, respect for patient dignity, respect for patient privacy and confidentiality, respect for the patient's right of informed consent, adherence to standards of professional integrity, and fulfillment of the responsibility to maintain professional competence.

      Aesthetics (specifically, medical aesthetics) is relevant to such fields as dermatology, plastic surgery, reconstructive surgery, and prosthetics.

      Political philosophy is relevant to the practice of medicine insofar as it may be concerned with such questions as: What is justice? How can social justice best be achieved? What constitutes justice with regard to access to health care? Does a just society provide universal access to health care? How can justice in health care delivery best be achieved? 

      The philosophy of language may be relevant to medicine insofar as it may explore styles of communication (among health care providers, between providers and patients, and between providers and the general public), and it may clarify issues related to cultural competence, workplace communication, and peer to peer communication. It may also be useful in determining how scientific problems can be presented more cogently and understandably to the general public. Medical semiotics (the recognition, interpretation, and evaluation of medical signs and symptoms) may also be an area of intersection between medicine and the philosophy of language.

      The philosophy of medicine may be relevant to the way in which physicians view the practice of medicine, and to the way in which they view themselves as practitioners. It may be concerned with such questions as: What is medicine? Is medicine an art or a science? What is good health? What is an acceptable quality of life, from a health standpoint? How can health care providers enable patients to attain their best possible quality of health? How can a patient best be recognized as a whole person, rather than as a collection of disease processes?