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Let me just state what I think are the three main issues at stake in this passage, leaving a fuller consideration of their implications to a later stage of this work.

Let me just state what I think are the three main issues at stake in this passage, leaving a fuller consideration of their implications to a later stage of this work.

1.  Reflective agents, as a price for their extra powers of reflection, fall prey to a new class of errors and liabilities, any one of which might be diagnosed as a ''reflective illusion'' or a ''delusion of reflection'' (DOR).  There is one type of DOR that is especially easy for reflective agents to fall into, and they must constantly monitor its swings in order to guard the integrity of their reflective processes against the variety of false images that it admits and the diversity of misleading pathways that it leads onto.  This DOR turns on thinking that objects of a nature to be reflected on by an agent must have a nature that is identical to the nature of the agent that reflects on them.

#<li value="1">Reflective agents, as a price for their extra powers of reflection, fall prey to a new class of errors and liabilities, any one of which might be diagnosed as a ''reflective illusion'' or a ''delusion of reflection'' (DOR).  There is one type of DOR that is especially easy for reflective agents to fall into, and they must constantly monitor its swings in order to guard the integrity of their reflective processes against the variety of false images that it admits and the diversity of misleading pathways that it leads onto.  This DOR turns on thinking that objects of a nature to be reflected on by an agent must have a nature that is identical to the nature of the agent that reflects on them.</li>

An agent acts under many different kinds of constraints, whether by choice of method, compulsion of nature, or the mere chance of looking outward in a given direction and henceforth taking up a fixed outlook.  The fact that one is constrained to reason in a particular manner, whether one is predisposed to cognitive, computational, conceptual, or creative terms, and whether one is restrained to finitary, imaginary, rational, or transcendental expressions, does not mean that one is bound to consider only the sorts of objects that fall into the corresponding lot.  It only forces the issue of just how literally or figuratively one is able to grasp the matter in view.

An agent acts under many different kinds of constraints, whether by choice of method, compulsion of nature, or the mere chance of looking outward in a given direction and henceforth taking up a fixed outlook.  The fact that one is constrained to reason in a particular manner, whether one is predisposed to cognitive, computational, conceptual, or creative terms, and whether one is restrained to finitary, imaginary, rational, or transcendental expressions, does not mean that one is bound to consider only the sorts of objects that fall into the corresponding lot.  It only forces the issue of just how literally or figuratively one is able to grasp the matter in view.

To imagine that the nature of the object is bound to be the same as the nature of the sign, or to think that the law that determines the object's matter has to be the same as the rule that codifies the agent's manner, are tantamount to special cases of those reflective illusions whose form of diagnosis I just outlined.  For example, it is the delusion of a purely cognitive and rational psychology, on seeing the necessity of proceeding in a cognitive and rational manner, to imagine that its subject is also purely cognitive and rational, and to think that this abstraction of the matter has any kind of coherence when considered against the integrity of its object.

To imagine that the nature of the object is bound to be the same as the nature of the sign, or to think that the law that determines the object's matter has to be the same as the rule that codifies the agent's manner, are tantamount to special cases of those reflective illusions whose form of diagnosis I just outlined.  For example, it is the delusion of a purely cognitive and rational psychology, on seeing the necessity of proceeding in a cognitive and rational manner, to imagine that its subject is also purely cognitive and rational, and to think that this abstraction of the matter has any kind of coherence when considered against the integrity of its object.

2.  The general rule of pragmatism to seek the difference that makes a difference has its corollaries in numerous principles of indifference.  Not every difference in the meantime makes a difference in the end.  That is, not every  difference of circumstance that momentarily impacts on the trajectory of a system nor every difference of eventuality that transiently develops within its course makes a difference in its ultimate result, and this is true no matter whether one considers the history of intertwined conduct and experience that belongs to a single agent or whether it pertains to a whole community of agents.  Furthermore, not every difference makes a difference of consequence with respect to every conception or purpose that seeks to include it under its "sum".  Finally, not every difference makes the same sort of difference with regard to each of the intellectual concepts or purported outcomes that it has a bearing on.

#<li value="2">The general rule of pragmatism to seek the difference that makes a difference has its corollaries in numerous principles of indifference.  Not every difference in the meantime makes a difference in the end.  That is, not every  difference of circumstance that momentarily impacts on the trajectory of a system nor every difference of eventuality that transiently develops within its course makes a difference in its ultimate result, and this is true no matter whether one considers the history of intertwined conduct and experience that belongs to a single agent or whether it pertains to a whole community of agents.  Furthermore, not every difference makes a difference of consequence with respect to every conception or purpose that seeks to include it under its "sum".  Finally, not every difference makes the same sort of difference with regard to each of the intellectual concepts or purported outcomes that it has a bearing on.</li>

To express the issue in a modern idiom, this is the question of whether a concept has a definition that is ''path-dependent'' or ''path-invariant'', that is, when the essence of that abstract conception is reduced to a construct that employs only operational terms.  It is because of this issue that most notions of much import, like mass, meaning, momentum, and number, are defined in terms of the appropriate equivalence classes and operationalized relative to their proper frames of reference.

To express the issue in a modern idiom, this is the question of whether a concept has a definition that is ''path-dependent'' or ''path-invariant'', that is, when the essence of that abstract conception is reduced to a construct that employs only operational terms.  It is because of this issue that most notions of much import, like mass, meaning, momentum, and number, are defined in terms of the appropriate equivalence classes and operationalized relative to their proper frames of reference.

3.  The persistent application of the pragmatic maxim, especially in mathematics, eventually brings it to bear on one rather ancient question.  The issue is over the reality of conceptual objects, including mathematical "objects" and Platonic "forms" or "ideas".  In this context, the adjective "real" means nothing other than "having properties", but the import of this "having" has to be grasped in the same moment of understanding that this old schematic of thought loads the verb "to have" with one of its strongest connotations, namely, that nothing has a property in the proper sense of the word unless it has that property in its own right, without regard to what anybody thinks about it.  In other words, to say that an object has a property is to say that it has that property independently, if not of necessity exclusively, of what anybody may think about the matter.  But what can it mean for one to say that a mathematical object is "real", that it has the properties that it has independently of what anybody thinks of it, when all that one has of this object are but signs of it, and when the only access that one has to this object is by means of thinking, a process of shuffling, sifting, and sorting through nothing more real or more ideal than signs in the mind?

#<li value="3">The persistent application of the pragmatic maxim, especially in mathematics, eventually brings it to bear on one rather ancient question.  The issue is over the reality of conceptual objects, including mathematical "objects" and Platonic "forms" or "ideas".  In this context, the adjective "real" means nothing other than "having properties", but the import of this "having" has to be grasped in the same moment of understanding that this old schematic of thought loads the verb "to have" with one of its strongest connotations, namely, that nothing has a property in the proper sense of the word unless it has that property in its own right, without regard to what anybody thinks about it.  In other words, to say that an object has a property is to say that it has that property independently, if not of necessity exclusively, of what anybody may think about the matter.  But what can it mean for one to say that a mathematical object is "real", that it has the properties that it has independently of what anybody thinks of it, when all that one has of this object are but signs of it, and when the only access that one has to this object is by means of thinking, a process of shuffling, sifting, and sorting through nothing more real or more ideal than signs in the mind?</li>

The acuteness of this question can be made clear if one pursues the accountability of the pragmatic maxim into higher orders of infinity.  Consider the number of "effects" that form the "whole" of a conception in PM₁, or else the number of "consequences" that fall under the "sum" in PM₂.  What happens when it is possible to conceive of an infinity of practical consequences as falling among the consequential effects or the effective consequences of an intellectual conception?  The point of this question is not to require that all of the items of practical bearing be surveyed in a single glance, that all of these effects and consequences be enumerated at once, but only that the cardinal number of conceivable practical bearings, or effects and consequences, be infinite.

The acuteness of this question can be made clear if one pursues the accountability of the pragmatic maxim into higher orders of infinity.  Consider the number of "effects" that form the "whole" of a conception in PM₁, or else the number of "consequences" that fall under the "sum" in PM₂.  What happens when it is possible to conceive of an infinity of practical consequences as falling among the consequential effects or the effective consequences of an intellectual conception?  The point of this question is not to require that all of the items of practical bearing be surveyed in a single glance, that all of these effects and consequences be enumerated at once, but only that the cardinal number of conceivable practical bearings, or effects and consequences, be infinite.