"For my own part, I would rather excel in knowledge of the highest secrets of philosophy than in arms." — Alexander the Great
"For my own part, I can do philosophy my own way." — Me the not Great
Philosophy is the generalised study of any thing. Things are of two kinds: real things and abstract things. Real things are also called as entities and abstract things are also called as ideas. Reality or physical world is the totality of entities. The world is the totality of all things. The totality of all abstract things is the realm of ideas or ideal world.
The branch of philosophy that studies the fundamental nature of reality is called Metaphysics. The goal of metaphysics is to find what things are entities (i.e. what really exists?) and what are their descriptions and relations with each other. Ontology is the branch of metaphysics that studies existence. Mereology is the sub-branch of ontology that studies part-whole relationships of entities.
Entities with many-to-one relation with other entities are called as compositions. A monobloc chair is composed of many plastic molecules. Many components form the one composition hence there is a many-to-one relation. Compositions have extra properties compared to the composing entities. Properties are the things that describe other things. A chair can be comforting but plastic molecules are not comforting like a chair. A composite entity is not fundamental as it can be reduced to its components either in description or in reality as the process of decomposition.
"Ontological simples" are fundamental entities that are not composed of anything so they cannot be further decomposed. They can compose other entities by interacting with other entities or simples.
Entities with one-to-one relation of their composition but not all properties are same among them are called constitutions (arrangments). An origami boat is a specfic constituion of a piece of paper. The boat can float on water but the plain paper cannot. New properties have emerged out of the constitution.
Metaphysics is often regarded as the first philosophy but I like to call it as low-level philosophy similar to low-level computer programming owing to the generality and depth of its questions. But who is doing these studies? It is the philosophical agents.
Agents are entities that are able to do something. Philosophical agents are special kind of agents that can experience reality and can reason. From now onwards I call philosophical agents as agents for brevity. You are an agent. I am an agent. Objects are entities other than agents. A chair is an object.
Abstract things are not existent in reality as entities but still they can be conceived and described by agents. For example, Spiderman, electric fields, point particles, line, etc.
Experience and reasoning are the processes through which an agent forms beliefs. A belief is the attitude (judgement) held by an agent about any description of properties of a thing.
A belief is qualified as true and be called a truth in two independent ways:
1. If it corresponds to reality (the correspondence theory of truth) -
To correspond means that the belief contains the actual descriptions of an entity as it is (such descriptions are called facts).
2. If it is reasoned coherently based on other truths or definitions (the coherence theory of truth) -
Reasoning is how an agent conceives any new belief (inferred beliefs) from already held beliefs.
Knowledge is truth that is justified (demonstrated or proven to be true). Epistemology (theory of knowledge) is the study of justification of truths. The goal of epistemology is to decide what beliefs are regarded as knowledge.
Epistemology is hence the branch of philosophy that focuses over what and how an agent can know about any thing either by experience (a posteriori) or by reasoning alone (a priori).
A posteriori knowledge - Beliefs that are concieved through experience and justified to be true only by observation or empirical evidence.
A priori knowledge - Beliefs that are concieved through reason and justified to be true without any need of observation or empirical evidence.
The branch of philosophy that studies coherent reasoning is called Logic. The goal of logic is to define the coherent rules of inference so that an agent can contruct coherent beliefs. Logic requires a formal system called logical language or syntax to express and organise beliefs. A belief expressed in some formal language is called a proposition. In a proposition, the subject is the description of a thing and the predicate is the description of properties of that thing. Based on the contents of a proposition, there are two kinds: Analytic and Synthetic.
In these propositions the predicate is the definition of the subject. They are a priori as they do not require empirical evidence. They do not necessarily conform to reality but are true because the definitive properties must not contradict each other (coherent truth). Examples:
1. “A quadrilateral has four sides.” (Quadrilaterals are defined as planar shapes which have four sides.)
2. “A sister is a female sibling.” (“Female siblings” and “sisters” are literally the same thing.)
In these propositions the predicate is not the definition of the subject, where properties other than definitive properties are claimed. They are contingent either on reason or observation:
Without using any sensory information or inferential experiment, the truth value of these claims cannot be decided. Hence they are true only by conforming to reality. Examples:
1. “It is raining."
2. “Water is H2O.”
Consider the proposition: "There is a hippopotamus in this matchbox." I can try to observe it but cannot find it, and then it can be argued that it is hidden because the inability to observe the hidden hippo does not make the claim false. But one can take the definition of a hippopotamus and a matchbox and reason that it is bigger than a matchbox, so it cannot be in this matchbox. Without observing a hippo, we can say that the claim is false. In a way, we claimed that “A hippo cannot be in a matchbox” to be true, which is synthetic but not a posteriori.
Synthetic A priori claims do not require direct experience or observation to be true. “For a triangle, the sum of the angles is 180 degrees.” This is not the definition of a triangle but it is still true without any direct observation. However, it is impossible to prove the angle sum property of triangles without having the notion of space. To define something, all sufficient properties must be met. After that, one can find necessary properties. Reason is needed but also some sufficient background experience (conceptual) and definitions.
All synthetic a priori claims are true in their conceptual framework, so the cases where the conceptual framework does not work, the synthetic a priori claims do not work also. For example, the angle sum property is true for triangles in Euclidean geometry only. Hence, a priori synthetic claims are true only coherently. For mathematicians all synthetic a priori claims are also analytic as such claims are derived out of axioms that are also analytic.
The initial beliefs from where reasoning starts are called premises and the inferred beliefs are called conclusions. An argument is a set of premises that justifies a conclusion. Hence arguments are a formal usage of reasoning to establish knowledge.
An argument is called deductive if the conclusion is already included in the premises but may not be obvious to an agent. Hence deductive argument "deduct" the required conclusion from the premises. They guarantee the truth of conclusion if the premises are true. Deductive arguments are useful in generating a-priori knowledge.
If the premises being true guarantees the conclusion also being true then the argument is called as a valid argument. Moreover if the premises are true then such a valid argument is called a sound argument, otherwise it is called an unsound argument. Invalid arguments do not guarantee the truth of conclusion hence are also called as deductive fallacies.
An argument is called ampliative if the conclusions are not deduced from the premises but rather involve new beliefs. Ampliative arguments are useful in generating a-posteriori knowledge. A non-deductive or ampliative argument is of two types: Abductive and Inductive.
A table highlighting different types of arguments, given by Peirce
| Abduction | Deduction | Induction |
| Case from Rule and Result | Result from Rule and Case | Rule from Case and Result |
| Rule (first principle): All the beans in this bag are white | Rule (first principle): All the beans in this bag are white | Case (hypothesis): These beans are from this bag |
| Result (conclusion): These beans are white | Case (hypothesis): These beans are from this bag | Result (conclusion): These beans are white |
| Case (hypothesis): These beans are from this bag. (The beans were taken out of this bag) | Result (conclusion): These beans are white. (These beans that we have now are white) | Rule (first principle) All the beans in this bag are white. (all the beans taken out will be white) |
Only deductive argument guarantees the truth of the conclusion. Abductive and inductive arguments are trial fits to make the deduction work. The idea of non-deductive or ampliative arguments introduces the concept of probability. The probable the conclusions, the stronger the ampliative argument. If the premises are in fact true then such strong arguments are called cogent arguments.
In abduction, we seek a probable case that makes the conclusion deductively true given the rule is true.
In induction, we seek a probable rule that makes the conclusion deductively true given the case is true.
In a way, abduction is about finding what probably happened (finding the case that fits the rule), and induction is about what probably will happen (as the rule dictates it). Deduction is finding what happens.
Relative to a deductive conclusion, the premises and argument involved are called its proof. For any synthetic a priori proposition, a proof can be demanded. Based on how we proceed to supply proofs, we find the following problem:
There are only three ways of absolutely completing an argument:
1. The circular argument - in which the proof of some proposition presupposes the truth of that very proposition. The problem is that it is not an argument but rather some elaborate rephrasing of the proposition to be proven as its proof.
2. The regressive argument - in which each proof requires a further proof, and so on. The problem is here the unending nature of such arguments.
3. The dogmatic argument - which rests on accepted propositions that are merely asserted rather than proven. Hence simply rejects to be an argument at all.
Popper suggested to accept this trilemma as unsolvable and construct knowledge by the method of conjecture and criticism.
But why an agent holds any belief and bother about arguments and justifications? The motivation to hold any belief is decided from the agent's values. Values are principles or standards that specify what is considered worthwhile or desirable. Axiology is the branch of philosophy that studies values. Analogous to how I called metaphysics as low-level philosophy, axiology can be comparatively called as high-level philosophy. Values are of two types: moral and non-moral.
Non-moral values are values that involve permissible actions. Moral values are values that involve obligatory or forbidden actions. Ethics is the study of moral values and Aesthetics is the study of non-moral values.
Moral agents are agents that can make choices based on what values they hold. Ethics is hence the study of an agent’s choices. In ethics, the goal is to find the right choices (what an agent is ought to do - moral principles). Moral choices can only be made when the agent is able to make any choice. Kant described this fact as "Ought implies can." Which means "If you are ought to do it then first you must be able to do it." Hence studying the agents themeselves is the key of any axiological study.
Ethics is contingent on choices (actions) and the ability to experience (suffering and pleasure). Both choice and experience are linked via reasoning. Reasoning converts experiences into beliefs and to chosen actions. But Choice, Reason and Experience are the fundamental aspects of sentient consciousness and ultimately contingent on the problem of how it emerges out of unconscious components (the Hard Problem of Consciousness). This is how axiology is contingent on epistemology and logic and that being contingent on metaphysics.
For aesthetics, the studies of cognition, psychology and humanities are the critical aspects as they really explains what and how much aesthetic value something holds for an agent.
The process of conjecture and criticism (Popper) when formalised is called science. In my view, science is the method of philosophy. The goal of science is to create a description of any system (the part of world that is being studied) by establishing its properties and the relations among its properties by reason, experience or both. At any instant such a description is called a scientific theory. Scientific knowledge can hence be defined as coherent true beliefs which withstand repeated scrutiny and revision. The apparently non-scientific fields like humanities and social studies (law, politics, history, etc.) do contain partial features of the scientific method if the goal is to describe and make inferences. Depending on the system, scientific studies are of two types: empirical and formal.
In case of formal science, (where the system is abstract but may also have some correspondence to reality) the theories are exact but still can be revised.
Mathematics, Statistics, Computer Science are all examples of formal science.
Formal sciences often work on exploring theories of abstract systems that may model some general features of natural systems.
Specifically we can define mathematics as the branch of formal science which deals with abstract systems that may have idealised features of real world or natural systems.
The axioms (fundamental premises that are assumed to be true) are set in such a way that the deductions lead to useful or at least interesting results (theorems) that are universal in application and deductively valid.
Theorems are the propositions deduced from axioms or already deduced theorems. The argument presented to verify the deduction of a theorem is called proof of that theorem. A system of axioms is called:
Complete: if every true theorem must have a proof then the system of axioms is complete.
Incomplete: If you can't prove some theorem to be true or false using some system of axioms, then that system of axioms is incomplete.
Decidable: if some algorithm (standard method) must exist so that whether the statement has followed from an axiom can be checked then the system is decidable.
Consistent: If it is free of contradiction. Being able to prove something but not its opposite (not converse but inverse).
Basically if I can prove some statements to be true from my axioms, I should not be able to prove the same statement to be false, otherwise my system of axioms is inconsistent.
In the case of empirical science, (where the system is a part of reality. For example: Physics, Biology, Social Science, etc.) the theory does not "exactly" describe the system but a model of the system which is an approximation of it. In empirical science every observation of a system may give us knowledge of some properties of the system. The model has to be developed such that it explains all the known properties. But such a model is not complete; it has to predict new observable properties. If such predictions are observed then such observations become evidence for the theory. If observations are contrary to the predictions then the theory is revised or discarded. This feature of theories in empirical sciences is called falsifiability. Falsification enables us to test if the theory is empirical in nature or not. This process has to be repeated in the search of finer theories and this process is called scientific research. This way different classes of philosophical questions gave birth to different fields of science.
How does observations of a system give us knowledge of its properties?
Observations are possible because we possess the ability to sense our environment. The raw outcomes of these senses form the experience.
Measurement is the process of transforming observations into objective, recordable and communicable description of any property of a system. A measured property is a constructed predicate:
a label we use to describe certain aspects of a system that consistently correlate with our sensory experiences. These labels gain meaning through interaction and comparison.
Every measured property is hence a record of a sensible effect arising from interaction of the observer with the system. Because of these interactions, measurements are inherently comparative.
Measurement is either categorical or numerical. Unlike categorical measurements (assigning a category value to a measurement),
in numerical measurements we have to define reference values to standardize measurements. These reference values are called units.
How to have trust in measurements?
The result of an experiment may have several measurements and it becomes naturally crucial to describe the quality of a measurement.
To quantify the quality of any measurement we must report both the best estimate and the uncertainty about it.
Precision is the degree of agreement or consistency (reproducibility) among multiple measurements of the same property. Lack of precision is termed as Uncertainty and numerically it is the degree of the spread of values we measured for a given property. More is the precision less is the uncertainty and vice versa.
An instrument measures a property numerically. Let’s say it shows 010.1040. Since here the 0 (left to 1 before decimal) is not a significant digit, we safely re-write it to be 10.1040. Why are we keeping the rightmost 0? Because it is significant. It is obvious that non-zero digits are significant but the significance of zeros between nonzero digits and trailing zeros have a little more for us to understand. These zeros are telling us that there is 0 contribution of that particular digit place (power of 10) and having such shows the finest possible measurement that can be made with that instrument, often called the least count which also serves as a measure of precision for the instrument. For the above example, the least count is 0.0001 because the measurement is possible upto the fourth decimal place.
How exactly digits and precision are related? In general, n number of digits can represent 10^n distinct values.
Hence adding an extra digit will enable us to represent ten times more than the previous. Consider that I measure something to be 2.01 and then later as 2.02
then I am uncertain about the digit at the third decimal place. Having least count improved from 0.01 to 0.001, I have an extra digit that can represent 10 values in the range between 2.01 and 2.02.
This makes my precision to increase 10 folds. Adding one more digit, will improve least count to 0.0001 and now I can represent 100 values in the same range which again 10 folds the precision.
As a rule, we can infer that in base m number system, adding an extra m-bit, increases the precision m-folds.
This is very interesting in the case of computers, where having one extra bit to represent numbers, increases precision by two times. So extra 1 Byte (8 bits) increases precision by 2^8=256 times.
Landauer found that you need a minimum amount of energy to cause thermal fluctuations somewhere in the electronic circuits
to destroy atleast one bit of information and as we discussed above it will lead to loss of precision by half.
Accuracy is the closeness of agreement between a measured value and a true or accepted value (precise results that most experiments have consistently given). Measurement error is the amount of inaccuracy and is defined as the difference between the measured value and the ‘true value’ of the thing being measured. Hence a measurement is only meaningful when we have enough knowledge of the errors and uncertainty involved. It is possible that people use the words “error” and “uncertainty” interchangeably but error values can be any real number (since it is a difference between two real numbers) and uncertainty is always a non-negative real number since it is a spread.
A measured result agrees with a theoretical prediction if the prediction lies within the range of experimental uncertainty. If two measured values have standard uncertainty ranges that overlap, then the measurements are said to be consistent (they agree). If the uncertainty ranges do not overlap, then the measurements are said to be discrepant (they do not agree).
Science requires experiments. Experiments give measurements. Recorded measurements are called data. Statistics is the formal science where we develop method for collection, analysis, interpretation, presentation, and organization of data. Probability theory is the mathematical framework for quantifying uncertainty. Statistics uses probability theory to make inferences. Inference is the process of drawing conclusions from incomplete, indirect, or uncertain information in any experimental data.
Probability - The mathematics of ignorance
In probability theory, an experiment is any procedure that yields a well-defined result. A trial is a single execution of the experiment.
The outcome of a trial is the specific result obtained, also called a sample point. The set of all possible outcomes is called the sample space.
An event is a subset of the sample space, consisting of one or more outcomes that satisfy a particular condition of interest.
To quantify the probability of an event say E, denoted as P(E) we can either:
1. Run real experiments (frequentist approach) - P(E) is the limiting ratio of the number of times the event E occurs to the number of trials say N, as the number of trials approaches infinity.
2. Or contruct a probability distribution model (theoretical approach, some may call it Bayesian)
where we define probabilities as fractions going from 0 (certainly not happening) to 1 (certainly happening) for each outcome (this is the probabilty distribution function that maps outcomes to a real number in the closed range from 0 to 1)
with the condition that the sum of all these probabilities has to be 1 since the sample space contain all possible outcomes
and it is certain that atleast one outcome out of all will happen in any trial.
P(E) now becomes the sum of all the probabilities of outcomes that belong to the subset of event E.
This theoretical approach is verified or tested against the observed probabilities and updated to capture the patterns in an experiement.
The best estimate is the expectation value of the probability distribution that models the data. The uncertainty is the standard deviation of the mean, also called standard error, which is the standard deviation divided by the square root of the number of measurements made. Standard deviation is the square root of variance. Variance is the expectation value of squares of data values minus the square of expectation value of data. Expectation value of a variable x with probability distribution P(x) is the probablity weighted average of all the values of x over the entire sample space.
It is commonly asked that while sciences have progressed then why philosophy has not made any progress? But my answer to this question in the defense of philosophy is that because of traditionally how philosophy is practicised and popularised, it came out as "subjective intellectual concerns to examine different beliefs" rather than a formal study of things. Though examining beliefs is infact should be called as formal study. Some people can argue that philosophical questions conventionally are such that they cannot be studied or answered via the scientific method and arguments are the only way to make progress in philosophy but the scientific method and research itself is a form of argumentation. If we stick to the "study of things" definition then any intellectual progress should be considered a part of philosophical progress. But since the meaning of progress is contigent on values of the agents then even sciences are not immune to such questions of "not making enough progress". In my view, the progress of philosophy resides in the variety of questions it can discover and the ways it compells us to refine our ways of experiencing and reason about our world.