Concepts are elements that live in the mind. They are used to build understanding. If we want to share this with others, we may use words or pictures. After an extensive explanation, it is still not certain whether this really succeeded. Words and pictures only describe the concept, they are a projection from another domain. As essential aspects can be lost, we have to trust that the concepts behind our words are the same for everyone. Confirmation can be obtained in a conversation.
Facts are the elements of observations. Measurable characteristics of objects in the physical world can be shared if well-defined procedures are used. Ultimately, every measuring procedure is based on counting. To find the gravitational constant, the time and height of a falling object must be measured. Both are determined by counting: time by the number of light wave cycles generated by cesium atoms during the fall, weight by the number of grams it takes to balance the object.
Numbers are concepts that we all have in the same way. Perhaps they are the only ones shared by all people of all times around the world. Two arms, five fingers, it’s clear what we mean. Everyone trusts that everyone else understands the meaning of a number. Numbers bridge observations with concepts as they live in both worlds.
Concepts that cannot be related to physical observations and ultimately to numbers in a universally accepted way are always the cause of discussions and misunderstandings. Examples are intelligence, health, peace and well-being. Physical objects such as a table, a pen, a house, a car or a road relate to concepts that may seem obvious. But for each of them, we occasionally disagree with each other. For example, the table of one can be the stool of the other. It should be realized that concepts are not fully covered by a word or an object and a single word can refer to several concepts.
In scientific discussions sharp definitions are often needed to clarify arguments and reasoning. At some point, however, concepts can shift, take on slightly different properties and it is necessary to change definitions, arguments and proofs. For example, the center of our universe changed over the centuries from the earth to the sun to undefined. In such situations, scientists must try to restore their mutual understanding. Ultimately, trust is needed that this point has been reached.
Let us summarize the above. Concepts live in the mind. Some of them, such as intelligence and well-being, only manifest in a few measurable features. Others are related to objects or phenomena, which in turn have measurable characteristics. Measurements are related to numbers. Ultimately, every measurement is based on counting. The resulting natural numbers have a unique property. They are concepts that are shared by everyone without the need for further explanation. Consequently, they are very useful in building confidence that other concepts, which are only partly manifested by measurable characteristics, are shared between people.
Numbers are concepts that live in the mind and clearly manifest in the physical world. They unite the two worlds in the same way for everyone. Numbers support confidence in the way in which these worlds are connected.