In the process of this entry I will be evaluating a particular strand of scientific thought known as scientific realism - or commonsense realism as it is sometimes called. This term refers to a family of positions which share a common belief in the real-world implications of scientific theories, a view that science informs us of the structure of the common reality we all inhabit, and which for the large part exists independent of what people say and think of it. Naturally, this interpretation of science satisfies a great number of scientists and laymen as it maintains continuity with tradition and adheres to a commonsense view of the world and our place within it. I am of the belief that its established orthodoxy in western society - implicit in the way science is reported - has a detrimental effect on our individual range of creative powers, since the bias is set against the relativist strands of reasoning which fuels the largely esoteric interpretations of post-modernism. Scientific realism has an aura of authenticity that stifles western imagination and puts an unnecessary obstacle in the way of man’s capacity for self-configuration (a topic to be discussed further in a future entry). But more importantly and to the point: scientific realism lacks sufficient proof, despite popular opinion to the contrary.
I wish to make this clear by using methodology as a demarcation criterion for science, mathematics and the external realm. This will not, I hasten to add, be a thorough investigation of the demarcation problem as it exists in the philosophy of science, as the present focus is limited to an epistemological evaluation of the methodologies at work in science and mathematics, rather than on the expansive implications of science as a field of study. This inquiry will bear on the demarcation problem only indirectly, in that it will provide a default demarcation of scientific reasoning which could be utilized in further refinements of the larger topic of science.
Both science and mathematics share a common methodological dependency on formal systems as a way of deriving theorems. formal systems were proposed as a way to exhibit every step of a proof explicitly, within one single, rigid framework, so that any mathematician or logician could check another’s work mechanically (Hofstadter p194). Their basic properties include rules of inference, axioms and theorems. The rules of inference delineate the boundaries of the system’s authority and are themselves unchallenged, in the same way Euclidean geometry can only be formally applied by first accepting his first four postulates uncritically. To the best of my knowledge rules of inferences are usually expressed in linguistic terms, as statements in common language. Foundational axioms are similar in that they are uncritically true but differ in that they are comprised of well-formed strings of symbols pertaining to the rules of inference, and are essentially free theorems. Foundational axioms are not always necessary for a formal system to work, for example in propositional calculus where a ‘fantasy rule’ is included in the rules of inference (see pg 183 hofstadter), but are common in pure mathematics. Theorems are produced within the formal system by stringing together the inducted symbols of the system in such a way as to comply with the rules of inference and foundational axiom(s). like axioms, they are formal constructs, but unlike axioms, they develop through the process of applying the rules of inference and are thus provable through this process. Together, rules of inference, axioms, and theorems comprise the basis for formalized analysis.
Calculus and Geometry are two obvious examples of formal systems but it is important to realize that formal systems are not restricted to these abstract disciplines but also include isomorphic correspondences between two different kinds of notations, for example in logic and grammar by using a propositional calculus which incorporates words and word-phrases into symbolic form, analyzes, and reverts them back into well-formed theorems. I used the term ‘isomorphic’ above, and this needs to be defined: an isomorphism is an information-preserving transformation wherein two complex structures can be mapped onto each other where there is a corresponding part, meanings that two parts play similar roles in their respective structures (Hofstadter pg 49). This concept of isomorphism will play an important part in my argument against scientific realism later in this entry.
For now, I would like to accomplish my original intent, to demarcate science from mathematics on the basis of method. As mentioned above, both science and mathematics inevitably depend upon some implementation of formal systems of analysis in their method, if only indirectly through the collaborative nature of their fields. Whereas mathematics is capable of being expressed intelligibly without the need to interpret beyond the confines of the formal system, as pure mathematics, science, insofar as it is independently coherent from mathematics, depends on isomorphic correspondence with an external realm. as such, the scientific method can be generally described as the external application of one or more formal systems, a quality Hofstadter classifies as ‘system-plus-interpretation’. This application results in a transference of information from the formal system to an ‘external realm’ which in the process must be interpreted, if only passively, by having a motivation for the particular isomorphism one is investigating (i.e. associating real-world concepts with mathematical symbols). In order to forego extraneous philosophical debate and preserve the issue at hand the external realm may be described as that which supercedes the domain of the formal systems. Both science and mathematics share a common dependence on formal structure that is distinguishable from the informal potentiality of the external realm. while the external realm supercedes the domain of these formal systems it does not necessarily exclude their content (the formalized theorems); thus, there exists, at the very least, the potential for overlap correspondences between the information of all three domains.
Mathematical method = formal system or system-plus-interpretation
Scientific method = system-plus-interpretation
External realm = informal and formal potentialities
A scientific theory insofar as it pertains to something external (which is the above-mentioned criteria) depends upon derived rules, rules which go beyond the jurisdiction of the formal system to interpret correspondences elsewhere (Hofstadter pg 194) The integral formality of the system is lost when rules are derived from outside of the system in such a way as is required for a scientific theory to be applied. One may argue that there is an additional formal system pertaining to the application of derived rules, a meta-theory beyond the confines of the original system, assuring the integrity of the proofs, but the derived rules to sustain the meta-theory may also be challenged, resulting in an infinite regress. Thus a scientific theory is always an approximation of the formalized proofs and lacks the jurisdictional authority in their external applications that axiomatic reasoning has within a well-defined formal system.
But this approximation or verisimilitude should not be considered justification for interpretations (such as scientific realism), primarily due to the fact that the verisimilitude is not meaningful in any qualitative sense (value-judgments), but only insofar as a 1:1 correspondence is discovered between quantitative aspects of, for example, water and its formal notation equivalent. Qualitative interpretations are dependent on our conceptualizations of the world, not the formalisms; they derive their value from outside the confines of axiomatic reasoning. the problem of engaging a natural world isomorphism is you apply real-world concepts which have built-in additional connotations that exceed the rule-governed meanings of the concepts within the exchange. these familiar meanings inspire us to assume beyond the boundaries set by the formal rules of inference, and corrupt the analysis.
When the raw data of a scientific experiment is compiled, it becomes necessary at some point to posit possible uses for the data that may extend beyond this demarcation of science. These are derived rule interpretations which possess measurably less authority than the raw data findings. Being only able to substantiate the quantitative what of natural phenomena within the constraints of the isomorphic exchange, the scientist still has the capacity to imagine scenarios for use applicable to the world we inhabit, and use these derived rule interpretations to link raw data with potential uses based on assumptions of how the universe works. So long as these transgressions result in further expansion of raw data that is self-consistent within the framework of the quantitative what it examines, derived rule interpretations can be useful, but still in themselves bear nothing more than a means to an end, with no intrinsic authority.
This is a crucial distinction, and I think its misunderstanding has been the source of most of the mystification we live with today. People simply assume that the derived interpretations of scientific theory that help expand our serviceable raw data are themselves within the boundaries of genuine science, and can be allotted self-consistency in the same way that asserting the boiling point of water is 100 degrees Celsius could be proven self-consistent, or that 2 + 2 = 4 is self-consistent. This is false. The untouchable rules of inference impart a self-consistent framework within which mathematical analysis can speak of indisputable truths (i.e. 2 + 2 = 4) but it is a fallacy to believe these provisional truths correspond with the unfiltered framework of the natural world. the indisputable nature of mathematical truths pertains only to the self-consistent framework of the formal system, demarcated by the invented rules of inference. Or to put it simply: there can never be a complete symmetrical correspondence between scientific fact and the natural world as such an isomorphism is incommensurable with the relationship of scientific method and the external realm as described above. in the resulting asymmetry of scientific method and the external realm no qualitative interpretations can be authenticated formally, rather they are believed on faith as guiding concepts to expand or correct the accumulation of raw data gained through the exchange. The qualitative meaning of the color red, for example, may be gleaned indirectly but since no sufficient formal system can prove it, it is open to interpretation and is therefore disputable. the notion of indisputable formalized truths is tautological (the formalized truths are true insofar as the system can be said to be true, which it cannot).
I distinguished sufficient formal systems in the above sentence for a reason. And this has to do with the revelation brought about to me through Hofstadter’s book regarding the multiplicity of systems imaginable, some of which transcend our steadfast bias for real-world usefulness. One has to arbitrarily impart a common base to a formal system which is provisional, and includes criteria such as consistency internal and external, and one that abides by basic tenets of logic and mathematics, to be enforced through the rules of inference. As Hofstadter attests, â€œconsistency is not a property of a formal system per se, but depends on the interpretation which is proposed for it.â€™ (pg 94) consistency imbues passive meaning, and is thus beyond the domain of the system. He goes on to explain:
“A system-plus-interpretation would be logically consistent just as long as no two of its theorems, when interpreted as statements, directly contradict each other; and mathematically consistent just as long as interpreted theorems do not violate mathematics; and physically consistent just as long as all its interpreted theorems are compatible with physical law; then comes biological consistency, and so on.” (Pg. 96). These consistency measures have to be pre-programmed into the formal system and are not intrinsic to them. This brings us to the provisional license of the foundational rules of inference of any formal system, which creates rather than discovers indisputable truths according to a provisional use of reason maintained by faith. It is these faith-based boundaries of the domain of the system that signifies its tautological quality, unless the criteria programmed into the system is inclusive of irrational inconsistencies, which is possible but not applicable to the integrity of science and mathematics.
This returns me to scientific realism, one of the most prominent of the derived rule interpretations because it presumes a complete isomorphism between exact science and the natural world. I consider it a fallacy for several reasons; first, it is impossible to formally prove completeness of the isomorphism; secondly, the desired information transfer between formalisms and natural phenomena can be done successfully without dependence on a value-judgment as to why. This second point leads me to advocate the instrumentalist position on science that claims theories are merely instruments to predict observations. Instrumentalism opposes the realist belief in a hidden structure inversely determining accurate scientific endeavor.
If we were to expand from an epistemological evaluation of method, to include the wider implications of science as a field of study, it would be essential to somehow integrate “both a general strategy and a complex social structure that carries out the strategy” (godfrey-smith pg 5). For as a field of study, science is dependent on interpretations - real-world analogies of formalized theorems - so as to encourage collaborative expansion of the field, and is thus dependent on an artful integration of formal knowledge, and what may be termed tacit knowledge, knowledge which supercedes objective explicitness, and which is potentially individualistic. While the explanatory power of theories are not an element of scientific reasoning they are essential in the social sense. Yet the social dimension includes tacit knowledge, knowledge hidden from ourselves that according to Michael Polanyi achieves comprehension through indwelling, as a result of our existential state. As he puts it: “we can know more than we can tell” and this relates to Wittgenstein’s discussions on seeing-aspects. How does science reconcile this tacit dimension with its own social structure? I will end with two quotations that indicate further the complications inherent with reconciling formal and informal potentialities as is intrinsic of science.
“If an ‘isomorphism’ is very simple (or very familiar), we are tempted to say that the meaning which it allows us to see is explicit. We see the meaning without seeing the isomorphism. The most blatant example is human language, where people often attribute meaning to words in themselves, without being in the slightest aware of the very complex isomorphism that imbues them with meanings.
Above I used the word isomorphism in quotes to indicate that it must be taken with a grain of salt. The symbolic processes which underlie the understanding of human language are so much more complex than the symbolic processes in typical formal systems, that, if we want to continue thinking of meaning as mediated by isomorphisms, we shall have to adopt a far more flexible conception of what isomorphisms can be than we have up until now. in my opinion, in fact, the key element in answering the question ‘what is consciousness?’ will be the unraveling of the nature of the isomorphism which underlies meaning.” (Hofstadter pg 82)
“The acceptance of scientific statements by laymen is based on authority, and this is true to nearly the same extent for scientists using results from branches of science other than their own. Scientists must rely heavily for their facts on the authority of fellow scientists.
This authority is enforced in an even more personal manner in the control exercised by scientists over the channels through which contributions are submitted to all other scientists. Only offerings that are deemed sufficiently plausible are accepted for publications in scientific journals, and what is rejected will be ignored by science such decisions are based on fundamental convictions about the nature of things and about the method which is therefore likely to yield results of scientific merit. These beliefs and the art of scientific inquiry based on them are hardly codified: they are, in the main, tacitly implied in the traditional pursuit of scientific inquiry.” (Polanyi 64)
And finally, my position in list form:
1) The formal system dependency of scientific and mathematical methods admit a certain priority to irrationalism in that all formal systems rely on invented rules of inference. thus their proofs are provisional when applied to an external reality.
2) The indisputable nature of scientific and mathematical proofs are tautological, and should be treated as such within philosophical investigations.
3) The scientific method can only predict natural phenomena (the quantitative what) and only with approximate accuracy. science cannot qualitatively explain the world we inhabit with any formal authority, by the very provisionary limitations imposed in its methodology.
4) Neither science nor mathematics bear a symmetrical (complete isomorphic) correspondence with the world we inhabit, and therefore are not interchangeable in analogous interpretations
5) Neither science nor mathematics bear a symmetrical (complete isomorphic) correspondence with one another, and therefore are not interchangeable in analogous interpretations
Hofstadter, Douglas R. Godel, Escher, Bach: an Eternal Golden Braid, Basic Books inc, 1999.
Polanyi, Michael. The Tacit Dimension, anchor books, 1967.
Godfrey-Smith, Peter. Theory and Reality.