What is Logic?
Edward Burger and Michael Starbird wrote a popular book on mathematics, The Heart of Mathematics: An Invitation to Effective Thinking. The title says it all, mathematics is an invitation to effective thinking. It is about ideas and their organization. Mathematics is built on logic. Euclid (mid 4th century), the great Greek scholar is called, ‘the father of geometry’. He built all geometry over 10 core logical principles.
“Let no one ignorant of geometry enter my doors”, those words were inscribed over the entrance to Plato’s Academy. Before you enter my academy, you must learn geometry because geometry is the gateway to attaining knowledge of the world. It teaches us how to think. It proceeds with logically valid arguments.
This importance of geometry continued down the centuries. The basic school curriculum had seven liberal arts: Logic, Geometry, Arithmetic, Astronomy, Grammar, Rhetoric and music. Every discipline starts with logic. Geometry, arithmetic, astronomy, grammar, rhetoric and music….they all must have a logical structure.
It has been said that philosophy has 4 branches:
1.Logic: The Study of rational argumentation
2.Epistemology: The Study of knowledge
3.Metaphysics: The study of reality
4. Axiology: The study of value judgments
A.Ethics: The study of value judgments regarding right and wrong for free human actions
B.Aesthetics: The study of value judgments concerning beauty.
Philosophy is not a waste of time. We should not think that philosophy is for people who have nothing productive to do. If you look at those 4 branches, in the first two branches, there is a lot of ‘talk’. But as you go towards the final branch there is a lot of ‘life’. Our lives are full of value judgments. What is right? What is wrong? Is this beautiful? Is this ugly? Yes, philosophy starts with so much ‘talk’, which often looks unprofitable and unproductive. But that is where you start before you reach more important areas like morality and ethics. Before Ethics, before Aesthetics, we have to start with logic.
After logic comes epistemology, the study of knowledge. All knowledge must be tested by logical validity. We come across the Three-Circle Technique in the study of validity. For the last two thousand years, we used those three circles as the gold standard for all knowledge. All information must flow from what you already knew. Mathematics is the only discipline that fulfils this criteria.
You will also notice that along the way, you must study metaphysics, the study of reality. We study about ‘truth tables’ in logic, because logic is necessary to discover truth and reality. The goal of logic is to discover what is rational to believe, what is truth. Reality must be established before we can talk about ethics and aesthetics.
Philosophy starts with logic and logic starts with three assumptions called the laws.
The law of identity
The law of non-contradiction
The law of excluded middle
Now who wrote these laws? My son Jacob is 9 years-old. He creates logic puzzles and throws them at people who come across his way. Recently he came to me and said, ‘Dad, I will give you a puzzle and you should give me an answer. There are four people on an island. They belong to one of two groups: Truth-tellers and Lie-tellers. One day the four people were having dinner. An outsider went to the island, and asked these four individuals, ‘How many of you belong to truth-tellers. Those four individuals gave 4 different answers: 0,1,2,3. Based on their answers, how many truth tellers are on the island? I said, ‘one’. Jacob with a smile on his face, said ‘You are correct,daddy’. I solved that puzzle using the law of non-contradiction. I asked my son, ‘Jacob, have you ever heard about the law of non-contradiction?’. He said, ‘no daddy, what is it?’ He never heard about the law of non-contradiction but he created a puzzle based on it. Why is that so? because we intuitively knew that if four people give you four different answers, only one of them could be right.
This child-like fascination with logic and mathematics led to many great discoveries in science. Henri Poincare (1854 – 1912) was considered one of the greatest mathematicians ever lived. He explored and contributed to almost every branch of mathematics. Using logic and mathematics, he formulated special relativity, independent of Albert Einstein. Poincare noted, “It is by logic that we prove, it is by intuition we invent”. Mathematical logic has such power to lead us to fascinating territories of physical reality not yet explored by scientists.
Logic and Mathematics
Where is Logic?
We tend to think ‘logic is for logicians: ‘I don’t have to worry about it. There is not much place for logic in my life’. But the reality is, we use logic in most of our daily activities. Words like technological, biological, sociological carry ‘logical’ with it. Why do you like your favorite politician? You might give a few reasons why you like that politician. You are trying to present a logical conclusion based on your reasons. My math teacher would give me a problem to solve. He would say, ‘explain step by step how you arrived at this solution. If you need, you can use logarithms.’ He was referring to Napier’s logarithms, which combine two Greek conceptions of mathematics, logos or logical reasoning and arithmos, or calculation.
Mathematics: We use logical thinking in mathematics. We take a problem and go in steps to solve that problem. It is a systematic way of thinking. According to logicism, all of mathematics is just complex logic. Logic is the reason for and the justification of mathematics. German mathematician and philosopher, Hermann Weyl (1885-1955) observed, “Logic is the hygiene the mathematician practices to keep his ideas healthy and strong”. Mathematician Ethan D.Bloch noted, ‘Logic is the framework upon which rigorous proofs are built’ (Proofs and Fundamentals, Ethan D.Bloch). We find the most powerful arguments in mathematics because nowhere else does the human mind invest in logic more intensely than in mathematics.
Science: Logic is also the backbone of scientific method. The concept of a physical universe, which obeys the laws of logic, has been extremely successful in the last five hundred years that it became the Central Doctrine of Scientific Method. From the experimental data to conclusions, we follow a logical order. Among others, it gave us microwaves, air planes, computers, satellites, vaccines and antibiotics.
Law: We use it in the court. Imagine a criminal trial. One witness said “I saw the defendant exactly at the location of the crime, around the time the crime was committed”. “Objection, your honor”, the defender’s attorney would stand up and say, ‘if the defendant was at the location of the crime, that does not establish that the defendant committed the crime. Correlation does not mean causation.” Logic has a central role in our legal disputes. The law demands that there must be a connection in“time, causation, or logic”.
Ethics: For so long it was thought logic had nothing to do with ethics. But, now philosophers have been using modal logic to investigate ethical obligations.
Epistemology: Epistemology is the study of knowledge. All knowledge must be tested by logical validity. We come across the Three-Circle Technique in the study of validity. For the last two thousand years, we used those three circles as the gold standard for all knowledge. All information must flow from what you already knew. Mathematics is the only discipline that fulfils this criteria.
Descartes followed the Euclidean approach to knowledge. He is noted for the invention of analytic geometry. All information must be deduced from indubitable truths that are available to us, such as the axioms of geometry. He saw the power of logic in mathematics and wondered where else we could go using it. In his 1637 book titled, ‘Discourse on the Method of Rightly Conducting the Reason to Find the Truth in the Sciences’ he wrote, “Those long chains of reasoning, so simple and easy, which enabled the geometricians to reach the most difficult demonstrations, made me wonder whether all things knowable to man might not fall into a similar logical sequence.” Descartes contends, using ‘long chains of reasoning’ we did great things in mathematics. Why can’t we apply the same approach to understand everything under the sun?
Metaphysics: Metaphysics is the branch of philosophy which examines the fundamental nature of reality. Mathematics helps us in linking the patterns inherent in the world around us because mathematics itself is the study of patterns!
Computer science: Computers are a product of logic. Computers won’t be computers without logic. What is a computer? Any electronic, high-speed data processing device that performs logical, arithmetic, or storage functions. It performs logical functions. Logic came first, then the computer. George Boole (1815 – 1864) was a English mathematician. In 1854 he wrote a great book titled, The Laws of Thought on which are founded the mathematical theories of logic and probabilities. Claude Shannon (1916-2001) was an American mathematician and electrical engineer. He is known as the father of information theory. He brought Boolean logic into computer technology. In 1948, he showed that electrical circuits could implement the rules of propositional logic and created the central design for electronic computers. It’s the Boolean logic which powers your computer!
As we enter the arena of artificial intelligence, logic leads us forward. Alan Turing’ famous computer was a machine made of logic: imaginary tape, arbitrary symbols. American mathematician Norbert Wiener (1894 – 1964) is considered the father of cybernetics, the science of communication as it related to living things and machines. He famously said, ‘Information is information, not matter or energy’. He linked his cybernetics with the theory of information developed by Claude Shannon.
Psychology: Paul Cisek in a paper published in 1999 in the Journal of Consciousness Studies, argued that computer metaphor provided mechanisms for unanswered questions perplexing neuroscientists in the areas like cognition, brain memory, and psychological phenomena. The computer has hardware and software. The brain is seen as the hardware and the mind as the software. Just like we formalized computer science with logic, symbols and equations, we can formalize psychological phenomena. Psychology literally means the study of the soul. If logic operates our minds, why not our souls?
Philosophy: As we have seen at the beginning of this chapter, logic is a branch of philosophy. Logic operates even in answering questions like the meaning and purpose of life. For example, Alvin Plantinga applied modal logic to ontological arguments.
Logic and metaphysics: Started together
What is the source of logic? From Greeks to Medieval theologians to modern mathematicians, everyone wondered about the origin of logic. For the theologian, God is the first logician. For the materialist, logic is an accident of nature.
Let us start with the Greeks. In their book about the history of logic, The Development of Logic, William Kneale and Martha Kneale called Plato, ‘the first great thinker in the field of the philosophy of logic.’
In The Republic, Plato develops a tripartite theory of the soul. Its three elements are nous (“reason”), epithumia (“passion”), and thumos (“spirit”). For Plato, logic is part of the soul. For modern materialists, logic is a product of physical matter. Plato says no. It is part of our immaterial soul. We look for a logical order in everything in and around us because it is part of our soul. For a while, logic stayed in the otherworldly Platonic realm. However, in the hands of Aristotle, logic was brought down to the materialistic world.
Medieval theologians extended Plato’s ideology into Christian theology. They nurtured the central doctrine of science: Nature is completely lawful. Medieval scholastic theology combined Christian theology and Aristotelian logic. St.Albert the great (1200 – 1280) was called the ‘Universal Doctor’ for his erudition in a wide range of disciplines such as philosophy, theology, botany, zoology, and paleontology. He went to the University of Paris (c.1241) where he became the first German master of theology and lectured on theology from 1245 to 1248. He was probably the most important person in the propagation of Aristotle’s logic in Europe. St.Thomas Acquinas was his student.
St.Albert wrote a massive paraphrase and commentary on all of Aristotle. Following Aristotle, his methodology for natural philosophy included observation and empiricism. He argued that logic should be the basis of study. The study of the natural world should lead to the glorification of God, beside ‘satisfying curiosity’.
We talk about the law of parsimony. We got it from William of Ockham. His most famous writing is his text book on logic, known as the Summa logicae. He gave us Occam’s logic or the law of parsimony, which is ideal in scientific investigations to this day. The Creator would not burden us with never-ending, unfruitful philosophical discussions. The simplest explanation with the fewest assumptions is most likely the correct one. Occam’s logic is derived from Christian theology.
Descartes was following in their footsteps. Professor Steven Gimbel explains in his work Einstein’s Jewish Science,
“Notice how Descartes, as a devout Catholic and a philosopher concerned with the nature of human knowledge, finds an approach for scientific reasoning that is perfectly cognate with this hierarchical arrangement. Catholic theology begins with large-scale, general, absolute truths bestowed on men infallibly from God, and all other derivative truths emerge from absolute certainty from a rigid hierarchical theological structure. In Descartes’ writings, we find a scientific method mirroring this, starting again with a general absolute truth provided by the Divine that gives rise to the absolute certainty of all further derived truths through a rigid and hierarchical logical arrangement. The discovery of both sacred and secular truths proceeds along similar plans, truths flowing downhill from undeniable first truths mediated by formal structures.”
For Descartes, God gave us the large-scale, general, and absolute truths which became the foundation for all other truths derived through logic.
Here we should talk about George Boole (1815-1864), who published the greatest work on logic since Aristotle: Boole’s Mathematical Analysis of Logic. He was born in a devout Christian family which emphasized studying natural philosophy in the service of God. His father John Boole constructed a telescope and placed it in his shop with the following invitation in the window: “Anyone who wishes to observe the works of God in a spirit of reverence is invited to come in and look through my telescope”.
In November 1849, George Boole became the first Professor of Mathematics at Queen’s College, Cork, where he taught the rest of his life. In 1854 he published An Investigation into the Laws of Thought, on Which are founded the Mathematical Theories of. Daniel J.Cohen writes in his book, Equations from God: Pure Mathematics and Victorian Faith,
“George Boole had a revelation at the age of seventeen that changed both his life and the course of Western philosophy. He spoke in spiritual and mystical terms about the insight gained in this epiphany, which forever separated modern logic from the logic of Aristotle and the Scholastics.
Despite the transference of his emotional insight into the mechanics of reasoning, Boole never forgot that the notion of a mathematical logic came in the form of what he regarded as a divine manifestation, and he hoped to return the favor by applying his new logic to the advantage of religious belief.
“The hope of his heart,” his wife Mary would later recall, had been “to work in the cause of true religion.” She continued, “Mathematics had never had more than a secondary interest for him; and even logic he cared for chiefly as a means of clearing the ground of doctrines imagined to be proved, by showing that the evidence on which they were supposed to rest had no tendency to prove them. But he had been endeavouring to give a more active and positive help than this to the cause of what he deemed pure religion.” The “father of pure mathematics”, as Bertrand Russell would later refer to Boole, had not been purely interested in mathematics, nor was his mathematics free of the “impurities” of extra disciplinary concerns, in particular, religious ones. The symbolic logic that is now an essential tool for secular philosophers and that forms the basis for dispassionate computers began in the mind of a warm-blooded, religiously concerned idealist.” (Equations from God: Pure Mathematics and Victorian Faith by Daniel J.Cohen)
For George Boole, mathematical logic came to us from God himself. Today’s secular philosophers and West Coast computer techies may not give any credit to God, but the symbolic logic they use began in the mind of a warm-blooded, Christian idealist.
John Venn (1834-1923)
The next great figure in the history of logic is John Venn. Using three circle figures, he gave diagrammatic representation to Boole’s logic. Do you know he was an ordained priest? He was the grandson of the founder of the Clapham Sect (a socially progressive religious movement) and son of the Secretary of the evangelical Church Missionary Society for Africa and the East. He was educated at Gonville and Caius College, Cambridge. His diagrams represent the structure of all 256 types of classical categorical syllogisms.
Logic and metaphysics: Divorced
While Christian thinkers such as George Boole and John Venn were enthralled in the study of logic as a gift of God, the secular thinkers were engaged in creating a materialistic basis for logic. Isaac Newton argued for a God who intervenes in his universe to help it run smoothly. Leibniz mocked Newton for turning God into a ‘cosmic plumber’. His ‘best of all possible worlds’ runs on the principle of sufficient reason, for everything God created must have reason for its existence and operation. From Newton and Leibniz we inherited a universe which runs on mechanical laws without any supernatural assistance.
This view was carried to its logical end in the hands of Scottish philosopher David Hume (1711-1776). Causation, maintenance, truth and morality must be anchored in experience. Logic is tied to experience. Nothing more. While recognizing its importance, Hume also warned us of the limitations of logic. His ethics are grounded on experience, not on logic. But for matters of natural philosophy, Hume relied on observation and logic. In his An Enquiry Concerning Human Understanding Hume asked, “if we take in our hand any volume: of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: For it can contain nothing but sophistry and illusion!”
Wait a minute, Mr.Hume. If I take your book in my hand and ask, ‘does it contain any abstract reasoning concerning quantity or number? Does it contain any experimental reasoning concerning matter of fact and existence.’ The answers are No and No. Can we call your assertions sophistry and illusion? This self-defeating posture becomes more evident in the modern offspring of David Hume, the logical positivists.
German philosopher Immaneul Kant (1724-1804) agreed with Hume about the limitations of logic. He said Kant ‘awakened him from his ‘dogmatic slumber’. Not surprisingly, we see a lot of Hume in Kant. He agrees with Hume that all of our knowledge arises from experience. But here is the glitch: it is not grounded in experience. He introduces two words, phenomena and noumena. Phenomena is our perception of the external world. Noumena is the external world as it really is. Science and mathematics are so successful because they deal only with the way things appear to us, not the way they really are in themselves. They deal with the world that can be experienced by our senses.
In his Critique of Pure Reason (1781), Kant explained the limitations of theoretical reason in science. We go to nature with pre-existing templates in our mind: time, space, substance, causality, action and reason. These templates help us to interpret the world. Mathematics is successful because it operates on these templates we impose on nature. Science is successful because it stays within the limits of our experience. Things like God, soul, free will, values and morality are beyond the bounds of sensory experience and of theoretical reason. We can still address those issues under ‘practical reason’.
Hume appealed to experience. But Kant recognized the variability of our experiences and made reason the guide to all scientific enquiry. Unlike Boole’s logic, Kant’s logic had no divine moorings. It is developed in the mind to make sense of the worldly experiences gained through senses.
French philosopher Auguste Comte (1798-1857) made Kantian philosophy as the basis of his positivism. He said, “There is no inquiry which is not finally reducible to a question of Numbers”. He coined the term sociology in 1838. He proposed that society progressed through three phases. It starts with the Theological phase, where all change is thought to be due to the actions of God. This Pandemic came due to anger of God, this abundance came due to Zeus, this earthquake is a punishment for sin etc. Then, society moves to the Metaphysical phase, where all change is due to unseen forces. Like Aristotle, we say there might be a final cause for the rain beside material cause, formal cause and efficient cause. In this phase, unseen forces are acceptable, but not God. Finally, society moves to the Scientific phase, where we get rid of both God and unseen forces and limit all inquiry to things experienced by our senses. Here we only talk about material causes, formal causes and efficient causes. Final causes are not our business. There is no place for teleology. Ask about ‘how’, not ‘why’.This final scientific phase is the positivist phase.
Comte made a religion around his philosophy. He established the Church of Positive Science with himself as the original High Priest of Humanity. It still has many followers in Brazil.
John Stuart Mill
Auguste Comte’s friend John Stuart Mill furthered empiricism among scientific circles. He agreed with David Hume that all science must take place in the domain of experience. He defined matter itself as “the permanent possibilities of sensation”. If you can’t feel it, it has no existence. He argued that every concept should be grounded in verifiable experience. This is not just limited to science. In his book A System of Logic, published in 1843, Mill demanded that every aspect of human life should be determined by empiricism.
Ernst Mach (1838-1916)
On the same lines, Austrian physicist and philosopher Ernst Mach wrote ‘An Analysis of Sensation’ with a mission to rid science of all metaphysical speculations. You must choose either science or metaphysics. You cannot have both. If you do science, you cannot do metaphysics and vice versa.
Austrian-British philosopher Ludwig Wittgenstein (1889 – 1951) worked primarily in logic and authored Tractatus Logico-Philosophicus (1921). His ideas strongly influenced a philosophical movement called logical positivism. Something can be ‘logical’ if it satisfies purely formal standards, and something can be ‘empirical’ if it satisfies purely experiential tests. In logical positivism, mathematics and logic are purely formal affairs. Their claims are necessarily true but empirically empty.
When Austrian philosopher of science, Karl Popper (1902 – 1994) was wondering about what distinguishes true science from pseudoscience, the logical positivists proposed verifiability as the criterion to establish the truth of science.
Wittgenstein posited that only meaningful sentences are those which are verifiable. He thought with this conclusion he answered all philosophical questions ever raised by humanity. If you say a chair, there must be a chair in space and time. It ain’t a chair if you cannot feel it whenever you want. If you talk about a table, there must be a table in space and time. Meaning comes from the correspondence to reality and verifiability. Things which are not verifiable through experience have no meaning. Philosophical problems are nothing but linguistic puzzles.
At Cambridge, Wittgenstein studied philosophy under Bertrand Russell. For many years, Russell entertained the possibility of the Platonic realm for mathematics but later gave it up due to the influence of his secular friends. John Gray writes in his Seven Types of Atheism, “Russell’s work in logic, in which he was a world-class pioneer, served a mystical impulse. For a time he seems to have believed that mathematical truths might exist in an eternal Platonic domain beyond time and the visible world. Until he was persuaded otherwise by George Santayana, he believed that values such as goodness and truth subsisted in the same ethereal realm. He spent much of his life searching for this heavenly domain and never found it.”
In his later years, Russell gave up on Plato and went with Aristotle. With Gottlob Frege (1848-1925), he believed and worked to reduce all mathematics to logic. Their years of hard work was nullified by Kurt Gödel’s incompleteness theorem, which proved that even mathematics is not self-sufficiently perfect to stand on its own legs.
Where is this secularized version of the Leibnizian universe heading? German mathematician David Hilbert (1862 – 1943) maintained that mathematical logic is nothing more than manipulation of symbols according to agreed upon formal rules. Being the most influential mathematician of his generation, he confidently announced, In der Mathematik gibt es kein Ignorabimus, meaning “In mathematics there is no we will not know”.
Through Entscheidungsproblem, he believed in arriving at proofs using a step-by-step deductive logic. But his grandiose project too was halted by Gödel’s incompleteness theorem.
Failures of Logical Positivism
Here, we should stop and look at the failures of logical positivism.
- It is self-contradictory
- It is self-defeating
- It is not consistent with the history of science
- It can not bridge induction and deduction
- It questions the realist view of science
- It questions the realist view of morality
- It gives rise to anti-science philosophies
- It undermines logic and mathematics
It is self-contradictory
Logical positivism starts with the assertion that ideas which are not empirically verifiable should have no place in scientific discourse. Yet, it embodies many unverifiable claims. John Stuart Mill in his A System of Logic (1843) declared that empiricism must shape every aspect of human life. He affirmed that the reality of nature is knowable and there is an underlying ‘uniformity of nature’. Sadly, those claims are unprovable by empiricism.
John Gray, a self-described atheist and author of “Straw Dogs” and “The Silence of Animals”, pointed that Mill, though rejected Christian monotheism, could not shake off its monotheistic way of thinking, such as ascribing reality, universality and uniformity to scientific method.
It is self-defeating
In Tractatus Logico-Philosophicus, Ludwig Wittgenstein professed that meaning comes only from correspondence to empirical reality and verifiability. He believed that he solved all philosophical questions ever asked by all philosophers. For many decades he failed to recognize that his own beliefs are not empirically verifiable. His bold declaration, “A statement is meaningless unless it can be verified” became the rallying cry of logical positivism. The irony of his self-defeating posture did not get lost on him. You can say, ‘“A statement is meaningless unless it can be verified”. Really? Can you verify what you just said?
It is not consistent with history of science
Logical Positivists thought that scientific progress comes through strict empiricism. To prove their point, they commissioned Thomas Kuhn (1922 – 1996), an American philosopher of science to investigate the progress of the scientific age. But what emerged was not what they expected. Kuhn published his findings in his landmark book titled The Structure of Scientific Revolutions (1962). He introduced the concept of paradigm. It says that every scientific theory contains a worldview within it. Scientists work within a paradigm built over cultural, social and economic forces. They adhere to a picture of reality within that paradigm. Even logic and mathematics operate within this paradigm.
Kuhn told logical positivists something they hated to listen: Observation is theory-laden. Empiricism cannot operate without pre-formed ‘unverifiable’ beliefs. Our best scientific theories must contain assumptions that are not themselves deducible from empirical evidence. These beliefs drive scientists what they should investigate and how they should interpret their investigations. These paradigms are not stable. They change over time in unpredictable ways.
LP advocates wanted to see the history of science as a gradual, cumulative progress, governed by logic and empiricism. But they found a different story, one of paradigm shifts, governed by the societies in which it is practised. Thus, the history of science does not support logical positivism. There are always synthetic a priori beliefs which are unprovable.
It cannot bridge induction and deduction
Science progresses through two processes: induction and deduction. In induction, we find new truths based on observations and experiments. In deduction, we go from already known true statements to new true statements using only logic. The problem is, how can we bridge induction and deduction? Logic cannot eliminate the gulf between the induction and deduction. Logic cannot provide justification for making universal assumptions based on our experience. For Kepler, God who made the universe also made the atoms. Whether you start with deduction or induction, they should sync with each other with no difficulty. Logical positivism does not have such justification to bridge the gulf between induction and deduction.
It questions the universalist, realist view of science
Ernst Mach (1838 – 1916), one of the founding fathers of logical positivism, proposed that empiricism is not about the reality of nature. We developed logic, mathematics, and scientific laws to explain our sensations of nature, not to explain nature itself. Thank you, Immanuel Kant. Not surprisingly, Marxists like Vladimir Lenin (1870 – 1924) attacked Ernst Mach, because if Marxism is only about explaining ‘our sensations’, not ‘the reality’ itself, it is meaningless to call it scientific materialism. What good is a philosophy which is untethered from reality?
From Plato, we inherited a view of science that is pursuit of knowledge, which is universal, necessary and certain. But, if man is the measure of all things, our experience is the only road to knowledge, built on local customs as Kuhn argued, then there is no justification to believe science as the way to universal knowledge.
If you add Darwinism to this mix, it will make you dizzy. If we fluctuate into existence randomly, if we interact with matter through sensations which fluctuate randomly in our brains, there is no rational justification to believe that we acquire universal, necessary and certain knowledge using science. We can never have any reason to believe our own thoughts about the experiments we perform. Not surprisingly, it is called Cognitive Instability.
If there is no justification for universal reality of knowledge, logical positivism would render all the laws of science to nonsense, because those laws are seen as universal laws of nature. Having seen this vulnerability, Karl Popper (1902 – 1994), an Austrian philosopher of science, rejected empirical verificationism as the criteria for science.
He proposed an alternative methodological standard called falsificationism. : Though a law cannot be verified an infinite number of times, it need only be falsified once to lose credibility.
It destroys the realist nature of morality
LP also destroys the realist nature of morality. Logical Positivism maintains that only statements which are valid are those which are empirically verifiable. In their view, moral statements like, ‘racism is evil’ have no cognitive meaning. If that is true, there is no point in condemning Richard Dawkins for making ‘transphobic’ comments or in criticizing James Watson for making racially insensitive remarks. Scientific community cannot make objective moral judgments on their colleagues if they take LP seriously.
It gave rise to anti-science philosophies
Logical Positivism was supposed to be the only road to objective knowledge. But, alas, it destroyed any claims to objective knowledge. As Kuhn suggested, if even the scientific method is built on local prejudices, what is the point of adhering to science as a source of objective knowledge? Consequently, the floodgates of relativism, postmodernism and deconstructivism were wide open in society as soon as logical positivism was established in our academia. Michel Foucault concluded that ‘knowledge’ is nothing but power relations. Jacques Derrida launched a relavisit assault on meaning and values. Logical positivism rather than establishing the scientific method as the only route to reality gave birth to Post-modernism, which questions science as the authority to truth.
It undermines logic and mathematics
In logical positivism, logic and mathematics became formal structures we impose on our perceptions and sensations of matter. They don’t have any independent existence in the Platonic sense. As David Hilbert suggested, If logic is nothing but a ‘formal apparatus’ agreed upon some mutually accepted rules, Postmodernism came to view rationality itself as an ideological device, not as a universal self-evident means by which human beings come into contact with reality and contribute to scientific progress.
If logic is only a temporary device, science is no more than a cultural paradigm, where does it take us? After Hitler took over Germany in 1933, Nazi sympathizer and philosopher Martin Heidegger was appointed rector of the University of Freiburg. In the inaugural address he declared that the essence of science is “the questioning standing of one’s ground in the midst of the constantly self-concealing totality of what is”. In What is Metaphysics, Heidgger writes that he wants to get rid of logical thinking, so that “every ideo of ‘logic’ dissolves in the whirl of a more basic questioning’.
In our own time, progressives view ‘logic’, ‘mathematics’, ‘objectivity’ as tools of White supremacy. They are not God’s gifts to humanity. They are just White man’s tools to suppress the colored people!
On 22 June 1936, when Moritz Schlick (1882 – 1936) was murdered on the steps of the staircase of the main building of the University of Vienna, he was vilified for embodying Jewish “logicality, mathematicality, (and) formalism”. That’s a tragic tribute to the founding father of logical positivism!
Logic and metaphysics: Come together
Contrary to the assertions of logical positivism, historians of science recognized the contributions of Christian theology to the birth of modern science. In 1936 Robert Merton (1910 – 2003), a founding father of modern sociology, studied the influence of Christianity on scientists who pioneered the Scientific Revolution. Under CUDOS (Communism, Universalism, Disinterestedness, Originality and Skepticism), he explained how Christian beliefs impacted the scientific careers of many members of the Royal Society of London.
The triumph of abstract mathematics in quantum physics and theory of relativity eroded the confidence of logical positivists who described logic and mathematics as nothing more than human constructions to explain empirical data. A sort of mysteriousness was accorded to logic and mathematics. Hungarian-American mathematical physicist Eugene Wigner (1902 – 1995), who received the Nobel Prize in Physics in 1963 explained this sentiment as , “The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.”
Kurt Gödel (1906 – 1978), who started his career as a logical positivist turned out to be its most significant pallbearer. His incompleteness theorem delivered “a sudden thunderbolt from the bluest skies” (Douglas Hofstadter) on the edifice of positivism.
In 1930, several influential mathematicians attended the Philosophical Conference in Konigsberg. On the final day of the conference, when Gödel presented his incompleteness theorem, it did not impress most attendees except a bright young mathematician named John von Neumann. Gödel’s theorem shocked him to the core. He later observed “the important point is that this is not a philosophical principle or a plausible intellectual attitude, but the result of a rigorous mathematical proof of an extremely sophisticated kind”…. “Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time….the subject of logic has certainly completely changed its nature and possibilities with Godel’s achievement.” Godel delivered a death knell to the ambitious projects of David Hilbert, Bertrand Russell, Gottlob Frege, and others.
Following Thomas Kuhn’s Structure, American philosopher of science, Norwood Russell Hanson (1924 – 1967) proposed that empiricism cannot stand on its own legs. It is people who see, not their eyes. People project their worldviews on their empiricism. Observation and theory must go hand in hand. He rejected the distinction between history of science and philosophy of science. They must walk together like a friendly couple.
American philosopher of Mathematics, Hilary Putnam (1926 – 2016) emphasized the importance of looking to non-scientific sources to understand metaphysics. Some philosophers have started to use modal logic to investigate ethical obligations.
Logic is taking us back to God
Christians understood that logic is a gift of God to humanity. During the Middle Ages, Chartres Cathedral was most renowned for the study of logic. Florentine artist Masaccio (1401 – 1428) painted the Holy Trinity for the Church of Santa Maria Novella in Florence. American art historian Mary McCarthy explains that this fresco, with its terrible logic, is like a proof in philosophy or mathematics, God the Father, with His unrelenting eyes, being the axiom from which everything else irrevocably flows. God is the source of our confidence in logic and mathematics. John Dewey rightly complained that, ‘Plato’s search for certainty is religion under another name. Great Christian mathematicians such as George Boole, George Cantor etc believed that the whole universe displays divine logic in its architecture. In his book Equations from God, historian of science Daniel J.Cohen tells the following story:
Charles Clarke recounted a typical story: “When I was a little boy, he (George Boole) took me, one Sunday afternoon, along the Greetwell Fields..The ministege Bells were ringing in for afternoon prayers and I told him that we should be late for Church. He smiled and told me that we were in Church, a church built by God himself and not by men, and asked me whether God could not be worshipped as well among the trees and in the fields as in houses made of stone?”
For Boole, mathematical laws were not products of the human mind but rather transcriptions of the mind of God. A favorite verse of Boole’s, “Forever O Lord, They word is settled in Heaven”, was for him a description of mathematical equations.
According to Mary Everest Boole, her husband “interpreted such expressions in the Psalms as ‘The word of God,’ ‘The testimonies of the Lord,’ etc., as meaning.. ‘all the laws of the universe’.
The Gospel of John starts with the words, ‘In the beginning was the Word, and the Word was with God, and the Word was God’. In the Greek version, it says, ‘In the beginning was the Logic, and the Logic was with God, and the Logic was God.’ In verse 14, it says, ‘the Logic became flesh and made his dwelling among us’. Thus, Jesus is the embodiment of divine logic. Not only immaterial realities such as logic, mathematics and moral laws do exist independent of matter, they are also interconnected in the person of God. I pray that you come to Lord Jesus Christ and invite him into your life as your Lord, Savior and Logician!
Paul Kattupalli MD